Kane scored in each half, finishing off a tearing Konrad Laimer run and converting a penalty, to raise his league tally to 35 goals in 31 games.
Kane is now six goals shy of Robert Lewandowski's all-time single-season Bundesliga goals record and has hit a personal best of 42 goals in all competitions this campaign.
The England captain's goals came either side of a Hugo Ekitike stunner, a day after the Frenchman became a permanent Frankfurt player.
Bayern veteran Thomas Mueller said his side was "in a good flow", saying since getting past Arsenal in the Champions League quarter-finals "we've been getting better, we've got a grip on things".
Before the match, Bayern coach Thomas Tuchel hit back at claims from the club's honorary president Uli Hoeness that the manager wanted to buy rather than develop players, saying the "absolutely baseless" allegation "could not be further from reality".
Despite the victory, which solidified Bayern's chances of a second-placed finish, both Laimer and defender Matthijs de Ligt were subbed off with injury just days before Tuesday's home Champions League semi-final against Real Madrid.
Dortmund, Germany's other Champions League semi-finalist, had a setback ahead of hosting Paris Saint-Germain on Wednesday, being thoroughly outclassed in a 4-1 defeat at Leipzig.
The undermanned visitors, missing captain Emre Can, Donyell Malen, Ian Maatsen and Sebastien Haller, dominated early, hitting the lead after 20 minutes through Jadon Sancho.
The England winger curled an excellent shot from the edge of the box into the top right corner to give Dortmund the lead.
Jolted to life, Leipzig equalised almost immediately, Lois Openda turning in a superb Xavi Simons cross for his 24th goal of the season.
Originally called offside, the goal was awarded after a VAR review and the video referee intervened again shortly after, overturning a penalty for Leipzig.
Leipzig then scored two quick-fire goals either side of half-time. Benjamin Sesko pounced on a rebound just before the break and Openda found Mohamed Simakan early in the second.
Christoph Baumgartner added a fourth with 10 minutes remaining to send Leipzig five points clear of Dortmund in fourth, the final guaranteed Champions League spot.
Dortmund goalie Gregor Kobel said "like so often this season, we gifted the opponent too many goals".
With three teams in the semi-finals of the top two European competitions, Germany may however snare a fifth Champions League spot.
Werder Bremen guaranteed first division football next season, winning 3-0 away at Augsburg to go 10 points clear of the drop with three games to play.
Goals from Romano Schmid and Olivier Deman and a Marvin Ducksch penalty took Bremen to a second-straight win for the first time since February.
Wolfsburg also took a major step towards securing top-flight football, scoring two goals in the final minutes to come from behind and beat a 10-man Freiburg 2-1.
In Saturday's late game, Bundesliga champions Bayer Leverkusen will try and keep their unbeaten record alive this season when they host third-placed Stuttgart.
No team in Bundesliga history has ever gone through a season unbeaten.
