Lelouch Lamperouge from Code Geass is known for this using his Magic Eyes to perform actions in such a way that makes it look like he performs miracles. Though he's actually at his best when he instead uses only his wits, brains and knowledge of people.
A good example of this is the "we are all Zero" gambit which is so outrageous it simply cannot fail, even though the logistics behind it make no sense.
Another example is the ending of the series. Lelouch becomes the biggest, most evil dictator in history, then has his Anti-Hero/Anti-Villain ally free the world by killing him. People unite against him! This allows people like Nunnally, Suzaku (as Zero), Kaguya (as AFN representative), Empress Tianzi and Ougi to help rebuild the world, with the side benefit of erasing Euphemia's Geass-incited machine gun massacre from public memory.
To say nothing of his on-the-spot plan to stop his parents' Assimilation Plot — he used his Geass on God and requested "Him" ("God" in ''Code Geass" collective unconsciousness of mankind, so technically, it has no definitive gender) to not stop the progression of time. God interpreted this as erasing his parents from existence, including his father, who, by this point, was the closest the CG-verse had to an immortal being.
In Fullmetal Alchemist, when Ed is doing his practical demonstration for his State Alchemist qualifications (at the age of 12, incidentally) he proceeds to not only create a spear from the ground without a circle in front of dozens of high ranking military officers, but to then threaten the Fuhrer King Bradley with it, just to prove a point. Bradley, curiously, takes it in stride, and even hints at his true nature as a homunculus, as well as what he's capable of, as he breaks Ed's spear without apparently flinching.
Later, this is subverted when Roy Mustang makes a joke about Bradley being a homunculus, and it is in fact taken very seriously. This is because, while Ed's threat was ultimately an empty one, Roy's questioning was taken very seriously because he's right, and too dangerously close to the truth for the corrupt higher-ups to let him go.
This is how Olivier Armstrong infiltrates the higher-ups. Bradley asked her what happened to General Raven, and she outright says (paraphrased) "I shoved him into wet cement. You should be grateful he's gone, he was so incompetent." This impresses him enough to let her in.
Isaac and Miria from Baccano! are able to fly under police radar despite being active criminals by virtue of the fact that the heists they pull off are so bizarre that the police don't want to be involved with them.
They once tried to steal a museum, after having infiltrated it as a pair of mummies. After discovering that the museum was too heavy to lift, they decided to make off with the entrance door so no-one else could enter the museum (the logic is impeccable, impeccable I say!). They get caught on camera in the process, and pose for the pictures. While still in their mummy disguises. The police, upon being delivered images of two mummies posing for the camera while carrying the entrance door to the city museum, do nothing.
They steal from Mafia by simply relying on the fact that nobody would be crazy enough to jump on them in clear daylight and run away laughing.
Their idea of disguises are so absurd and flamboyant that they're able to hide in plain sight. Their heists appear so bereft of criminal planning or malicious intent that they're taken for performance artists, perhaps pulling some dada stunts. Some of the pictures presented as 'evidence' of their crimes involve them possing with locals!
Akira Takizawa from Eden of the East pulls this off when the Japanese-English language barrier (as well as his own weirdness) leads him to conclude that the best way deal with the police officer questioning him is to drop his trousers and flash her. Far from arresting him on the spot for indecent exposure, the cop thanks him and lets him off the hook (leading many viewers to conclude that Akira's "Johnny" is so amazing that it has mind control powers).
Then there's the time when he flashed a random businessman on the street, then says something that causes the guy to laugh, then hand over his pants. Charm Person, indeed. And this happens in the first episode. It builds up from there.
Ayame from Fruits Basket. In a flashback to his school days, the only reason he was able to keep his long hair was he came up with an outrageous story about him being royalty and wouldn't shut up about it. And that's nothing to what he went on about as class president when some classmates got caught in the red light district.
In the final episode of Black Lagoon, Ms. Balalaika shows up at the home of a powerful Yakuza leader (under 24/7 police surveillance, no less), asks to see his gun and proceeds shoot him with it. Afterwards, she exits, shows a fake diplomatic passport to the policemen who came to investigate the gunshots, and then casually gets into her limo and drives away before anyone can figure out what happened.
Naruto gives the example of a diplomat attacking under a flag of truce in order to kidnap a three year old girl to steal her eyes. Her father catches him in the act and kills him on the spot. So what do his comrades do? Deny the kidnapping attempt ever happened, brand her father a murderer, and demand his execution and have his body turned over to them for confirmation, in order to obtain the set of eyes they couldn't get from the child! It would have worked but for a Heroic Sacrifice on the part of the man's twin brother, and is the source of a considerable amount of angst. They have yet to pay for this action in any way. This status was Lampshaded during a negotiation, and promptly dismissed.
Why Konoha was so willing to go along with this at the time is unclear, seeing as they had won the war that the diplomats had been ostensibly sent to negotiate an end to.
Most likely that Konoha was recovering from the lossof the 4th Hokage due to the Kyubi.
When up against what amounts to the God of the world, Naruto decides it's time to pulled out "That Jutsu". And "That Jutsu" is...the Reverse Harem No Jutsu. And the worse part is it actually works to a degree, letting Naruto get in a good hit.
Invoked in-story in Honoo No Alpen Rose. The music genius Leonhardt "Leon" Aschenbach is being pressured by Those Wacky Nazis alongside his best friend (and the female lead) Jeudi, and must write an "Austrian Symphony" as well as use it in a concert. What does he do? He accepts to play... but then he uses said concert to actually escape from Austria alongside Jeudi (via turning the lights off in a critical moment and then rushing out with the hep of his and Jeudi's entourage) AND give them the musical finger via inserting the melody of an anti-Nazi song (written by Jeudi's Disappeared Dad as the "hymm" of La Résistance) in his "Austrian symphony".And he gets away with it cleanly.
In Bleach, the Big Bad kicks off his on-screen run by having one of the True Companions executed on trumped-up charges, repeatedly has the time of execution changed, has his Dragon running around acting like the Big Bad (because he's so Obviously Evil), and in general mucks around with everyone. How? Nobody expected that one really likeable captain to fake his own death, nevermind kill off the Soul Society's government (and nearly his own lieutenant) or have been up to this shit for over a century. Even the Author didn't see it coming.
In Assassination Classroom, Koro-sensei, Nagisa, and Kayano free Kataoka from her guilt complex by forcibly teaching her blackmailer how to swim. They basically get away with abduction and imprisonment by dressing up as fish creatures, kidnapping the blackmailer in the middle of the night (bed included) while she slept, and dropping her off, bed and all, at their secret mountainside pool. When she wakes up, the blackmailer is fully convinced by their disguises and the location that she's merely dreaming and unwittingly goes along with their plan.