The 1969 Mets. The team had a history of poor performance, and limped off the line in 1969 with an 18-23 record, then suddenly turned things around to crush all competition and become the first expansion team to win the World Series. In the film Oh, God!, George Burns, as God, states that his only miracle since the Red Sea had been the 1969 Mets.
The 2004 Red Sox in the 2004 ALCS against their hated rivals, the Yankees. They were down 0-3 in the series and it was Game 4 with one inning from being swept, they flipped the switch and got dangerous. It wasn't just one person, it was EVERYBODY from Keith Foulke to Derek Lowe to Bill Mueller to Dave Roberts...heck, even Curtis Leskanic!! Everybody got dangerous and the team overcame the deficit and won the ALCS en route to their first World Series title in 86 years.
During Super Bowl XXII, the Redskins were being completely outplayed by the favored Broncos, all with starting QB Doug Williams playing injured after being sacked in the first quarter. With the Redskins down 10, and all momentum in the Broncos' favor, Williams and the Redskins exploded for 35 points in the 2nd quarter, where even little used RB Timmy Smith reeled off a 58-yard TD run, while Williams threw all 4 of his own TD passes to three different receivers, all the while showing exactly why he and the Redskins belonged in the Super Bowl in the first place.
A similar feat would be achieved in Super Bowl XLIII, when the underdog Saints team found themselves down 10-0 to Peyton Manning and the Colts early in the game, only to outscore the Colts 37-7 over the rest of the game, including an interception returned for a game-clinching touchdown at the end of the game, also known as the point where Peyton Manning was guaranteed to lead his team from the brink of defeat if necessary, as he had done in almost every game that season. This and the aforementioned Super Bowl XXII were the only two times in NFL history that a team was able to rebound from a double digit deficit to win the game.
During their 2008 season, the Arizona Cardinals, playing in the then-notoriously weak NFC West division, were statistically guaranteed entry into the playoffs despite a mediocre record. Meanwhile the New England Patriots, despite a superior record of 11-5 and a 47-7 victory over the Cardinals in week 16, didn't make the playoffsnote The Miami Dolphins and Baltimore Ravens also went 11-5, and due to both having better records within their respective conferences qualified over the Patriots, making the Pats the first 11-5 team to miss the playoffs since 1985. The Cardinals were harshly criticized, with some even suggesting the rules be rewritten to prevent such a thing from happening again. Then the Cardinals got dangerous, won their last game of the regular season and all three of their playoff games by convincing margins, made the Super Bowl for the first time in the franchise's history, and came within a couple minutes of winning.
The 2010 Stanley Cup Quarterfinals. The number 1 seed Washington Capitals and the highest scoring team in the league against the number 8 "we barely made it to the playoffs" Montreal Canadiens. With the Capitals up 3-1 by game 5, there was an obvious winner. And that's when Jaroslav Halak got dangerous. Over the next three games, Halak faced 134 shots... and let in a total of three. For reference, most games, a goaltender will face around thirty shots. Halak faced an average of almost 45, and in one game stopped 53... and won. His goaltending has been largely attributed as the reason for one of the biggest upsets in NHL history.
The 1980 USA Olympic hockey team. They trailed in every single game they played in the tournament and did not lose a single one; tied the first game against Sweden and won every game after that. And of course that included the "Miracle on Ice" against the overwhelming favorites from the Soviet Union, where, yes Al Michaels, we could believe in miracles.