The humble Swiss team, who qualified for the last 16 at the expense of Manchester United after a momentous victory against the English side, had St Jakob-Park rocking again as substitute Valentin Stocker scored an 86th-minute winner.
Basel, who often play in front of only a few thousand fans in the Swiss league, hit the woodwork twice, then survived long periods of Bayern pressure before snatching victory.
While Basel's players milked the crowd's acclaim, Bayern trooped off thinking the unthinkable.
The Bavarians, who have slipped to third in the Bundesliga after a dip in form, missed the injured Bastian Schweinsteiger, decribed by coach Jupp Heynckes before the game as the brains of the team and in the same class as Barcelona's Xavi and Andres Iniesta.
Basel's win was a rare boost for Swiss football following the national team's failure to qualify for Euro 2012 and the problems facing their domestic game.
"We are very happy but we know we're only halfway there," said Basel's coach Heiko Vogel, who has only been in charge since November and spent nine years working with Bayern's youth divisions where he coached some of the present Bayern team.
"The recipe is very simple, we have an unbelievable will to win and great understanding. We have fun every day during training and that shows."
Goalkeeper Yann Sommer, whose saves kept Bayern in the game, said: "We have beaten maybe the best two teams in the world, it's amazing what happened here, we believed in ourselves. Bayern played well but luck was on our side."
"If Ribery's chances had gone in it would have been a different game," said Heynckes.
Bayern had two let-offs in quick succession, firstly when Aleksandar Dragovic's header was pushed away by Neuer, hit the inside of the post and bounced out.
The woodwork saved Bayern three minutes later when Alexander Frei, Switzerland's record international goalscorer, crashed a shot against the crossbar from 12 metres after a Park Joo Ho's slide-rule pass found him free in the middle.
