Harry Potter is extremely popular everywhere. It might be more popular in Japan than in its native UK and is only second to the United States. Japan is even getting its own Wizarding World theme park.
The series of children's novels A Series of Unfortunate Events became popular in Canada long before they were in the U.S., and they were significantly more popular there.
A historical use would be the immense popularity of the Arabian Nights in Europe and America after they were first translated. While not hugely unpopular in the Middle East, the tales that came into Western knowledge were often not of massive importance and were actually looked down upon at several points in history. The popularity of translations, however, soared through the roof, having a huge influence on European and American writers and accumulating devoted fans and even fan societies.
France was the only place where Philip K. Dick achieved much fame as something more than a cult writer, until the last few years of his life. Possibly because the themes of his stories tended to dovetail with the ideas of then-current French postmodernist philosophy.
Stanislaw Lem is hugely popular in German intellectualist circles and his works are even part of many philosophy lessons at universities.
Creator/Enid Blytons "St Clares" series is hugely popular in Germany and has spawned TWENTY-ONE ghostwritten sequels which were published under the name Enid Blyton, as well as three movies. Mallory Towers is also extremely popular and has 12 ghostwritten sequels.
Jennings, a series of humorous English children's books set in a boarding school, were fairly successful in their native country but were (and are) overshadowed by the more famous Just William stories by Richmal Crompton. They became hugely popular in Norway under the name Stompa. The Norwegian translations of the books spawned four feature films and a radio sitcom series in the fifties and sixties. Reruns of the radio episodes are still being broadcast regularly, by popular demand.
Given that there's been manga, anime and video game adaptations of the Australian fantasy book series Deltora Quest, it must be mighty popular over there.
It became extremely popular in Sweden in the 1970s; so much that their national non-commercial TV made a film of the first half of Fellowship of the Ring (it was pretty bad, suffering from too much cheap blue-screen technology). Interestingly, the trilogy had already been translated in 1958 but spent the 1960s in relative obscurity.
It became insanely popular in the late 90's in Russia, spurring a huge fan subculture and lots of fanfics, several of which were printed. Nick Perumov, for example, debuted with one of those. Although this subculture has mostly faded through the last decade, Tolkien-themed LARP's are still among those with the most participants.
The Canadian novel Anne of Green Gables is very popular in Japan. There's even an anime based on it. The touristy areas of the real province of Prince Edward Island tend to have signs written in Japanese underneath English ones, also.
Frances Gordon (calling herself 'Bridget Wood') wrote a series of fantasy novels about psychic Celts and animal rape. It's mostly porn, gore and Gorn with a generous helping of bestiality. In the Netherlands, the (badly) translated books were marketed as YA and became one of the most popular fantasy series for teens for a while. Gordon even dedicated one of the books to her teenaged Dutch readers at one point.
Warrior Cats seems to be much more popular in America than in its authors' home country of Britain, though that may be because the publisher is based in the US. There have actually been more books released in foreign languages than there have in the British editions. The books are quite popular in Taiwan, which translates the books rather quickly - the books come with trading cards there, and there's an official fanclub, which gives them exclusive merchandise such as mugs. The series is also pretty popular in Germany, which has an official message board, is also translating the books at a rapid pace, has audiobooks, and is the only country outside the US (not counting a single day-long event in the UK) that has had the author tour there.
Frank McCourt's autobiography Angela's Ashes was better received in America (where it won a Pulitzer) than in Ireland, undoubtedly due to its less than glowing depiction of Limerick. Ironically, much of its international popularity was likely thanks to the late 90's surge in Hibernophilia.
Austrian author Thomas Brezina is quite well known in his home country, but extremely popular in China (especially The Tiger Team), where he managed not only to get into the Top 10, but at some point was the Top Ten — yep, all of the ten most popular books were his ones.
Israeli humourist Ephraim Kishon, while relatively well known in his home country, was (and to some extent still is) a huge name in Germany.
Czech writer Milan Kundera is hugely popular in Mexican intellectual circles. The Unbearable Lightness of Being is rather common in some high school curricula over there. However, he is also hugely popular, widely discussed and part of high school curricula in the Czech Republic. A scandal revealing his collaboration with the communist regime's secret police before he emigrated somewhat destroyed his reputation, though.
British writer Anthony Horowitz's Alex Rider series is popular in the United States (in a similar manner to the James Bond books as Alex Rider is basically a teen Bond) despite its British context, and spawned many imitations by writers from America and other countries.
In 1872, a British author named "Ouida" (Marie Louise de la Ramee) published a book called A Dog of Flanders. It's a sentimental Tear Jerker set in impoverished rural Flanders about a boy and his dog. It faded from memory rather fast and is now quite obscure in the Western world... but a Japanese diplomat loved it, brought it back to his home country, and now it's considered a classic of Western children's literature there. The novel even draws Japanese tourists to Belgium (and Antwerp in particular), where they are moved to tears by the cathedral (it has to do with the infamous ending), leading Belgians to wonder what the heck is going on. (This also fits under "Anime and Manga," since the anime adaptations are part of the reason the story is so popular.)* Like Anne of Green Gables, this is also insanely popular among the Japanese to the point of having not one, but THREE anime adaptations (one 52 episode anime, one 26 episode anime, and a movie)! Heck, it's had fans since 1908.
While he may have lived there and based many of his books in Rhode Island, a surprising amount of Rhode Islanders have no idea who H.P. Lovecraft is.
France really loves Lovecraft and his Cthulhu Mythos (and other similar authors, to a degree). A 3-Door Stopper omnibus of his entire collected writings is perrennially reprinted since the 80's at least, and at any one time several publishers have a number of short story collections in print; for at least a decade (before Lovecraft's renewed popularity and the advent of Project Gutenberg) it was easier to find his books in French bookstores than in American ones. There's even a publisher, Nouvelles editions Oswald (NeO for short), specializing in late 19th/early-to-mid 20th century pulp authors, which runs collections of the works of Clark Ashton Smith and Robert E. Howard (also Lord Dunsany and Edgar Rice Burroughs) that are long out of print (or rarely reprinted) in English. Strangely enough, this can be at least partly traced to the high popularity of the Call of Cthulhu role-playing in the hexagon (see the Tabletop Games section), which made French geeks curious about the rest of Anglo-Saxon pulp literature.
The Australian novel Tomorrow, When the War Began was selected by the Swedish government in 2000 as one of the books most likely to inspire a love of reading in young people, and financed its translation and distribution to every school-age child in Sweden. This trope was likewise inverted with the cold reception that the books received when they were released in America.
German philosopher Oswald Spengler'sThe Decline of the West is popular among a small group of people (but consider that nowadays few people read Spengler in general) in Russia who like him because he predicted that in the future, a new culture might develop in Russia and bring the country (probably) to greatness.
Les Misérables was extremely popular among Confederate soldiers during the American Civil War, due to their identifying with the book's doomed rebellion. Some even took to calling themselves "Lee's Miserables".
Brazilian author Paulo Coelho is quite popular in most places that are not Brazil.
In America and Britain 'The Second Coming' is by far the most famous work by William Butler Yeats. In his native Ireland it is no more famous than any of his other poems (possibly less so in fact as poems like 'Easter 1916' are more likely to be studied in school due to their historical content.)
The Moomin series, written by Finnish author Tove Jansson, is highly popular in Japan, and has had several anime adaptations there. It is also a staple of children's reading in Russia.
The self-help book The Servant is hugely popular in Brazil. In the US, it sold 200.000, but in Brazil it sold 2.4 millions and it was in best-seller lists for at least two years.
Edgar Allan Poe was initially much more popular in France than in America, partially due to unlicensed translations of his work by French poets like Charles Baudelaire. Of course, he eventually achieved widespread popularity in both countries (and elsewhere), but only after his death.
Gone with the Wind (the book version; the movie is banned) has found a huge, huge audience in North Korea. Apparently, it's because North Koreans can relate to the themes of struggling to survive in the face of war, hunger, and deprivation.
James Fenimore Cooper, not least due to Mark Twain repeatedly panning him, has somewhat gone out of fashion in his native United States, but in Europe he still is regarded as one of the more important American authors. In the former Eastern bloc, academic study of American Lit contained and contains an amount of analysis and teaching about Cooper that astonishes many Americans. During his lifetime, Cooper was lionized in France (where he e. g. wrote The Last of the Mohicans) and had a huge influence on European and specifically French writers, e. g. Balzac and Dumas Père.
David Gordon, a pretty unknown American writer, wrote a book called The Serialist in 2010. After a Japanese translation came out, he started winning Japanese literary awards. It became the #1 best-selling book there, titled "Niryuu Shousetsuka'', which means "Second-Rate Novelist". It got so big that they even made a movie adaptation. They invited him over for the screening and he was mobbed. He became so popular, that his Japanese publishers released the Japanese translation of his next book a month before its planned English release date. Supposedly, he became popular because of the book's perspective on women. The best part is, he was already an otaku!