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.
Medical examinations showed the 30-year-old right-back has lymphoma and he will now have several months of treatment.
Borussia Dortmund needed a win to lift the title but ended up drawing 2-2 with Mainz.
His recent Bundesliga scoring form has now spilled over to the World Cup and 'The Gap', as he is know due to a space between his front teeth, is being hailed as Germany's improbable hero.
Japan scored a thrilling 2-1 win over favourites Geramany in Group E.
The Baverians sealed the title with a 3-1 win over rivals Borussia Dortmund and with three games to spare.
Poulsen expressed confidence that the team will bounce back under new coach Domenico Tedesco.