Basel, inspired by 18-year-old Xherdan Shaquiri and backed by a raucous crowd, led 1-0 with a beautifully-crafted Alexander Frei goal until Bayern's Thomas Muller went down in a seemingly innocuous tangle with Benjamin Huggel in the 56th minute.
The midfielder played the ball but the referee pointed to the spot amid Basel's disbelief in an incident which may raise questions about the value of the two officials behind the goal, being used on an experimental basis in the competition.
Oblivious to the deafening jeers, the Germany midfielder known affectionately as Schweini buried the spot-kick and added insult to injury by creeping in at the far post to score an 89th minute winner after being left unmarked at a free-kick.
Basel coach Thorsten Fink, a former Bayern Munich player, chose not to criticise the decision. "I didn't have a good view of it but there were two handball incidents where I though we could have had penalties," he told reporters.
"I'm totally disappointed with this result."
MORE POWER
On a night when Bayern's hard-tackling Dutch midfielder Mark van Bommel committed only one foul, visiting coach Louis van Gaal preferred to praise his own substitutions which included bringing on Mario Gomez at half-time for Hamit Altintop.
"The changes gave us more power, we put more pressure on them in the second half and we scored the two goals," said the flamboyant Dutchman.
After a miserable month that saw Switzerland beaten 3-1 at home by England in a Euro 2012 qualifier and all three of their teams beaten in European competition two weeks ago, Swiss football needed a boost and for a while looked like getting one.
In the 17th minute Basel broke from defence, Marco Streller cleverly shielded the ball and flicked it to Switzerland's record scorer Frei who curled it into the top corner.
The goal was the cue for teenage winger Shaquiri, who scored a superb goal against England, to come to life.
He tormented the Bayern defence, created a buzz every time he got the ball and forced Jorg Butt to make difficult saves with powerful shots either side of half-time.
"We played really well, we fought and we represented Swiss football very well," said Fink.
Bayern, who have only eight points from six Bundesliga games and were missing the injured Franck Ribery and Arjen Robben, had plenty of possession but few chances until the penalty.
The last half hour saw chances at both ends before Bayern snatched the winner to top the group on six points with four goals scored - only one less than their Bundesliga tally.
AS Roma and CFR Cluj have three points after the Italians beat the Romanians 2-1 in Rome. Basel, who like Bayern won a domestic double last season, are still without a point.
