Kane bags brace as Bayern end Dortmund's title hopes
Bayern trailed at half-time after Nico Schlotterbeck headed Dortmund in front but Kane scored a tap-in and a penalty.
Dortmund levelled through Daniel Svensson but Kimmich was in the right place to blast in from close range with three minutes remaining to leave Bayern as runaway favourites for the title.
The result all but sews up another Bundesliga crown for the Bavarian giants with only 10 matchdays remaining, while continuing Dortmund's rough week, just days after their Champions League exit at Atalanta.
Dortmund have now lost just twice in 24 league matches this season, both times to Bayern.
Kane's fourth successive Bundesliga brace took him to 30 goals from 24 games this season.
Bayern had played two fewer matches in the past 11 days but Dortmund looked the fresher early on as they chased a victory to move them five points behind the leaders.
With 18 minutes gone, Schlotterbeck caught Josip Stanisic on the shin with his studs. Despite Bayern demanding the Germany defender see red, Schlotterbeck was shown a yellow card.
The centre-back was in the thick of the action again just eight minutes later when he rose high to head in a Daniel Svensson corner and put Dortmund in front.
Dortmund kept up the energy after the break but Bayern levelled things up against the run of play in simple fashion.
Kimmich chipped the ball over the Dortmund defence, onto the head of Serge Gnabry who squared for Kane to score in the 54th minute.
With just over 20 minutes remaining, Stanisic tussled with Schlotterbeck again and the Bayern player went down after the defender clipped his trailing leg, bringing Kane to the spot.
The England captain converted to give Bayern the lead. Kane now has nine goals in his past five Bundesliga games, four of which have come from the penalty spot.
Dortmund pushed forward in response and Marcel Sabitzer floated in a cross which Svensson swept home to give them hope on 83 minutes.
Not content to play for a draw, Bayern sought the goal which would all but secure them the league title.
That moment came as Kimmich did remarkably well to adjust his body and hammer in the decisive strike with his left foot after Dortmund had failed to adequately clear a Michael Olise cross in the 87th minute.
Leverkusen grab point
Former Liverpool defender Jarell Quansah scored with two minutes remaining to snatch Bayer Leverkusen a 1-1 home draw with Mainz.
Leverkusen trailed after an excellent Sheraldo Becker goal but Quansah squeezed the ball in at the near post late on.
The point moved Leverkusen, who have a game in hand to be played at Hamburg on Wednesday, three behind fourth-placed Stuttgart.
Leverkusen looked leggy after a hard-fought scoreless Champions League draw with Olympiacos midweek, which sent them through to the last 16 where they will face Arsenal.
Moments before half-time, Leverkusen right-back Arthur was stretchered off with what appeared to be a serious knee injury.
Elsewhere, St Pauli's resurgence continued with a 1-0 win at high-flying Hoffenheim, with Mathias Pereira Lage's goal just before half-time the difference.
Pauli have won three of their past four to climb out of the relegation spots, while third-placed Hoffenheim have picked up just four points in their past four games.
Borussia Moenchengladbach's Kevin Diks converted a penalty four minutes into stoppage time to take his side to a crucial 1-0 win over Union Berlin.
Gladbach's first win since early January took them three points clear of the relegation playoff spot.
Fellow battlers Werder Bremen also picked up a crucial victory, winning 2-0 at home to last-placed Heidenheim to move out of the automatic relegation placings.
Double winners in 2004, Bremen had picked up just four points in 13 games since their last win in early November, but won thanks to a Jovan Milosevic strike and a late Heidenheim own goal.