After winning the first-leg 3-0, Bayern could afford to sit on their lead but they looked the more likely to break the deadlock in the first half.
Kane felt he should have been awarded an early penalty after tangling with Leverkusen's Jonathan Tah, before the England striker forced a smart stop from goalkeeper Lukas Hradecky.
Michael Olise whistled a shot just past the post and then Kingsley Coman went close as the Bavarians enjoyed a bright opening 30. Conversely, Patrik Schick saw two headers fly narrowly wide as it was honours even at the break.
Seven minutes after the interval, Kane was on hand to tuck home Joshua Kimmich's free-kick as the striker capitalised on some dreadful defending. Schick spurned a glorious opportunity 22 minutes from time but he could only fire straight at keeper Jonas Urbig from close range.
The tie was well and truly dead when the former Tottenham striker neatly set up Alphonso Davies, who doubled Bayern's lead from 12 yards out in the 71st minute.
Jamal Musiala then fizzed an effort off the crossbar late on but it made little difference as Vincent Kompany's side won 5-0 on aggregate to see off their rivals at the last-16 stage on Tuesday at BayArena. Bayern will face Inter in the quarter-finals next month after the Italian champions saw off Feyenoord elsewhere on Tuesday.