BATE, who beat Lille by the same score in their opener, went ahead with a 23rd minute strike from Aleksandr Pavlov and added late goals through Vitali Rodionov and Renan Bressan as Bayern's nine-game winning streak in all competitions ended.
Bayern's Franck Ribery, who had earlier hit the crossbar, briefly cut the deficit in stoppage time but they could not stop 20,000 fans celebrating wildly in the Dinamo stadium.
Last season's Champions League finalists had dominated the first half and also hit the woodwork through Toni Kroos but were left to scramble forward in the second period, fearful of the Belarussian side's potent counter-attacks.
The result leaves the group wide open with BATE top on six points after their first ever home win in the competition and Bayern on three with Valencia, who beat Lille 2-0.
"It was a very difficult match but we fought until the end," said BATE winger Aleksandr Hleb, who spent seven years in the Bundesliga. "This is a sensational result for us. We will celebrate a bit but we need to keep working hard."
"Bayern are still favourites in our group. We are taking it one match at a time," said the former Barcelona player.
Bayern coach Jupp Heynckes, aiming to break a club record with a 10th straight win in all competitions, started with Javi Martinez instead of Bastian Schweinsteiger.
However, the Spaniard, who cost 40 million euros in Bayern's biggest close-season transfer, showed little initiative and was taken off early in the second half.
The Bavarians displayed only glimpses of the attacking game that saw them win all their matches this term until Tuesday.
"I knew this would be a very tough opponent and in such games you have to take the lead," said Heynckes. "We hit the post and bar and we had some bad luck."
"We constantly piled on the pressure but we have to give credit to our opponents. They are a very well organised team, especially in defence, and they showed it again tonight."
The hosts were not intimidated by their more illustrious opponents and Denis Polyakov forced a reflex save from Bayern goalkeeper Manuel Neuer after seven minutes.
BATE CELEBRATE
Bayern should have opened their account minutes later when Kroos snatched the ball from Dmitri Likhtarovich outside the box and rounded the keeper but he hit the post from five metres.
Instead it was BATE who celebrated first after soaking up mounting pressure from the Bavarians, with Pavlov taking full advantage of team-mate Aleksandr Volodko's mis-kick to rifle in from 10 metres.
Bayern, top of the Bundesliga with six wins from six games, instantly tried to bounce back but headers by Thomas Muller and Mario Mandzukic were smothered by keeper Andrei Gorbunov.
It was the hosts who came closest to another goal when Hleb backheeled into the path of Polyakov whose shot stretched Neuer.
Pavlov came close again midway through the second half, firing over the bar after another
