England captain Kane, who joined for a league record 100 million euros last week to solve Bayern's scoring problems, slotted their second goal in the 74th minute after setting up Leroy Sane for a fourth-minute opener.
Despite a stellar scoring record he failed to win any domestic or international titles with Spurs.
Bayern did not disclose the transfer fee, which media reports have put at around 100 million euros ($109 million). That would be a Bundesliga record.
Should it be completed, Kane's move would be the most expensive transfer in Bundesliga history with an expected fee of about 100 million euros.
A game billed as a crucial six-pointer in the race to secure Champions League qualification was effectively all over before the clock had reached 10 minutes as fifth-placed Tottenham collapsed like a pack of cards.
England's skipper struck a penalty past Italy keeper Gianluigi Donnarumma shortly before halftime to make it 2-0 and take his tally to 54, one more than Wayne Rooney.
Nottingham Forest beat Southampton to move out of the relegation zone.
The Gunners maintained their 100 per cent home record. They have 40 points from 15 games.
Having been a teammate of Kane at Tottenham for the best part of a decade, the 35-year-old was quick to reach out to the striker who was crestfallen at the final whistle.
Kane had scans on Wednesday but England manager Southgate said there was no serious damage.