American Football itself might not have survived past the 1910s due to the bloodiness of the sport (18 players died in 1905) had the forward pass not been legalized, thus opening up the game and preventing the 22-man dogpiles that led to the brutal injuries suffered prior to it.
Buffalo Bills QB Ryan Fitzpatrick stepped in as a starter last season and was mediocre. This year, he grew a sweet beard and he has been one of the top Fantasy Football quarterbacks, even if his team isn't winning.
In what is probably the most literal application of this trope, the Chicago Bears, Bulls, White Sox, and Blackhawks all won their championships under a coach or manager who had facial hair. Steve Rosenbloom says it best.
Cricket. Before the introduction of round-arm, and later over-arm bowling, balls were gently tossed underarm, and before the introduction of the LBW rule batters often blocked or left a very high proportion of shots, these two things combined to make the sport... somewhat less than thrilling for spectators.
But the sport really grew its beard in the late 19th century, a huge, black beard attached to the chin of W.G Grace, who single-handedly introduced the concept of modern batting.
It's difficult to remember now, but...there was a time that the Miami Heat basketball team was flatly atrocious. Then they went out and got a certain coach named Riley. It would be several years before a title, but the team became a contender almost immediately. A few years after their title, they did this AGAIN by signing LeBron James and still having enough salary cap room for the rest of the team.
The signing of NHL legend Mark Messier by the New York Rangers in 1991 signaled that the Rangers were serious about winning.
In the early and middle 2000's, superstar quarterback Peyton Manning was known for being the unchallenged best player in the regular season but sucking in the playoffs. In the 2006 AFC Championship, the dynastic powerhouse New England Patriots were leading 21-3 at halftime and everyone assumed this would be another Patriots-Colts Curb-Stomp Battle. Peyton grew the beard by engineering a huge comeback to win 38-34, then led his team to a smooth 31-17 victory over the Bears in the Super Bowl. Unfortunately, his teams' performances after that run are very similar to what they were before, but that game was enough to silence those who said he wasn't good enough to win a ring.
In 2010, San Francisco Giants closer Brian Wilson has done this literally and figuratively. His growth of a full beard has coincided with his evolution from a pitcher who would give Giants fans heart attacks every 9th inning into one of baseball's elite closers.
The Tampa Bay Buccaneers started life as a franchise by losing their first 26 games. In 1979, just their fourth year of existence, they went to the NFC Championship game behind the arm of prodigal quarterback Doug Williams. They spent most of the next two decades as a perpetual laughingstock, so much so that when Bo Jackson was drafted by the team he decided to play baseball instead. In the late '90s they hired Tony Dungy and quickly became a defensive powerhouse, but failed to win a championship. In 2002 they grew yet another beard, hiring Jon Gruden, who immediately took the team to the Super Bowl, defeating his old team, the Oakland Raiders.
The Tampa Bay Devil Rays. Ever since the team was founded in 1999, they had been the biggest laughingstock in the Major Leagues, finishing in last place in all but one season. Then, after the 2007 season, they exorcised the "Devil" from their name, becoming just the "Rays." The very next season, they won the American League pennant, and ever since, they've been a force to be reckoned with in a division that had previously been dominated by the Yankees and the Red Sox.
Tennis. Before the 80's tennis was a boring, dull sport to both play and watch. The very strict rules and coaches did not help. Then came the 80's. Antipathetic and borderline insane players such as John McEnroe and Ivan Lendl arrived. They threw their rackets, insulted and defied the coaches (to the point that the coaches were scared of them), and wore short pants. The quality and ratings of the sport increased immeasurably.
A slower Growing the Beard took place for the UFC. In the early days, it was very nearly fights with no rules, had no weight classes, and saw fighters going through many fights in one night during tournaments. Starting at UFC 12, weight classes were introduced, and more rules were adopted up to UFC 28, which was the first sanctioned event.
There were arguably two distinct beard-growing moments. The first was when Zuffa took control of the organisation. The second was the legendary bout between Forrest Griffin and Stephan Bonnar at the end of the first season of The Ultimate Fighter.
The Washington Capitals hockey team, there have been times in recent history when the team just sucks, even with star player Alexander Ovechkin. But then, in the middle of the 2007-08 season, when they seemed to be heading for another disappointing season, they fired then coach Glen Hanlon and hired Bruce Boudreau. Then suddenly, the Capitals managed to make a huge turnaround and ended up winning their division and reach the playoff in that same season. Since then, they had continued to dominate their division and are one of the big contenders to win the Stanley Cup. They even managed to win the franchise's very first President's Trophy (Given to the team with the best regular season record in the league) in the 2009-10 season.
Hiring coach Bill Parcells in almost any capacity is an automatic grown beard for almost any team.
Hiring Billy Martin to manage was arguably the baseball equivalent of growing a beard. Besides his notoriously five separate stints with the New York Yankees (with back to back American League pennants in 1976-77 and the World Championship in the latter), Martin took three other teams (the 1969 Minnesota Twins, the 1972 Detroit Tigers, and the 1981 Oakland Athletics) to the playoffs. Martin even helped the 1974 Texas Rangers to an 84-76 record (good for second place) after they had two consecutive 100+ loss seasons. Martin's greatest weakness however besides his combative personality and alcoholism, was his tendency to burn out young pitchers.
The Immaculate Reception is generally considered the Growing the Beard moment among Pittsburgh Steelers fans. Prior to this, the Steelers were considered a joke, with only five winning seasons in 40 years. After the reception, the Steelers have gone on to appear in seven a record Eight Super Bowls, winning six of them, and are now considered one of sports greatest franchises.
Similarly, "The Catch" by Dwight Clark was the moment the San Francisco 49ers went from plucky underdog to dominant team of the 1980s and 1990s.