For Bayern it was a measure of revenge for BATE’s 3-1 win in Belarus in taking Group F honours with 13 points while BATE had already guaranteed third place and a spot in the Europa League Round of 32 with six points.
Mario Gomez gave Bayern the lead after 22 minutes but the hosts were reduced to 10 men when defender Jerome Boateng was sent off after 51 minutes.
But Thomas Müller doubled their lead in the 54th minute and Xherdan Shaqiri effectively ended the match as a contest by making it 3-0 after 66 minutes.
BATE also finished the game with 10 men as Denis Polyakov was given a second warning for diving in the box in the 69th minute – David Alaba made it 4-0 in the 83rd minute before Yegor Filipenko scored a consolation goal for Borisov in the 89th minute.
BATE had the first chance of the game after 11 minutes with a low shot on goal from Aretm Kontesevoi, but Bayern keeper Manuel Neuer it.
Bayern grabbed the lead in the 22nd minute as Shaqiri fought his way through the left side and crossed into the box where Gomez poked the ball past BATE keeper Andrei Gorbunov for his first Champions League goal of the season and 21st of his career.
Bayern were close to doubling the lead in the 39th minute with Toni Kroos turning and unleashing a blast from 19 yards which Gorbunov did well batting out of danger.
Bayern went a man down in the 51st minute as Boateng was shown red for his foul on Kontsevoi.
But that didn’t matter as Munich made it 2-0 in the 54th minute with Müller volleying in Bastian Schweinsteiger’s header pass off Kroos’s long pass from the left side.
Gomez and Shaqiri swapped roles on Bayern’s third goal in the 66th minute as Gomez chipped a pass from the left side into the box where the diminutive Shaqiri (1.69m) headed home from five metres out to make it 3-0.
The visitors night deteriorated as Polyakov was booked a second time for overacting.
Alaba got on the score sheet in the 83rd minute as the Austrian international ran onto Ribery’s corner and blasted a left footed shot into the lower left corner to make it 4-0.
BATE grabbed a consolation goal in the 89th minute with Filipenko beating Neuer from about 15 metres out.