It was Kane who opened the scoring after 55 minutes, with a well-placed effort to beat Leipzig goalkeeper Janis Blaswich.
The visitors started the second-half brightly, as Benjamin Šeško levelled the game in the 70th minute.
Bayern looked set to endure a fourth competitive match without a win, before the England captain came up with the goods in the 91st minute to seal a victory at the Allianz Arena.
Kane's brace ensured Bayer Leverkusen were not 10 points clear at the end of Gameweek 23, by closing the gap to eight.
RB Leipzig remain in fifth position, as they let the opportunity to lift themselves into the top four ahead of Borussia Dortmund pass them by.
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Harry Kane stops the rot and spares Tuchel’s blushes
Robert Lewandowski holds the Bundesliga record for most goals scored in a single Bundesliga season with a remarkable tally of 41, beating former holder and Bayern legend Gerd Müller’s haul of 40.
Kane may well be on track to surpass both of them, having netted 27 goals in 23 games with eleven games still to be played.
The most impressive thing about Kane’s form is that despite the problems Bayern have in terms of injured absentees, defensive frailty and midfield mishaps, Kane has managed to dig his side out of a difficult situation by doing his part and finding the back of net.
Which happens to be quite literally more often than not, propelling Bayern forward despite the issues the club is facing at this moment in time.
Kane may well rue the chance to add yet another matchball to his already sizable collection as the opportunity to get his hat-trick evaded him narrowly.
His longest goal drought spanned just two games in January. Bayern lost one of those games at home to Werder Bremen and won the other with a single goal against 1.FC Union Berlin.
This, among other things, highlights the fact that Bayern struggle to score when Kane is not the one putting the ball in the back of the net.
Bayern’s turbulent season has already resulted in the club deciding to part ways with Tuchel at the end of the season.
It cannot be underestimated just how worse the club’s position could have been without Kane’s goals (27), which is nearly half of Bayern’s total for the season (63).
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Rose and Leipzig remain outside of the top four places
Third-placed VfB Stuttgart failed to beat relegation-threatened 1.FC Köln at home on Saturday afternoon.
The Swabians’ draw provided an opening for Leipzig to try and cement their position in the Champions League positions.
But, Kane’s late winner put that hope to bed and now Rose must go back to the drawing board ahead of a visit to the VfL Bochum’s cauldron on the Ruhr.
A win would have seen Leipzig leapfrog Borussia Dortmund in fourth, at least temporarily, as Die Schwarzgelben are still to play.
Edin Terzić's side now have the opportunity to put themselves four points clear if they beat TSG Hoffenheim at home.
Leipzig have shown worrying signs on a regular basis since the turn of the year, evidenced by the fact that they have won two of seven Bundesliga outings.
In Europe, they were timid and lacked bravery against an injury-plagued Real Madrid, and repeated that once again in similar circumstances against Bayern.
Bayern themselves know Bochum are no pushovers at home, and will be far from an opportunity for an easy three points for Rose's side.
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Despite the win, Bayern face an uphill battle to keep-up
It is a cliché, but Leverkusen’s win over 1.FSV Mainz 05 was a quintessential title challenger’s victory.
Hard work met luck when Robert Andrich’s effort slipped out of Robin Zentner’s hands, resulting in Xabi Alonso’s side continuing their impressive charge.
The season is not over and Bayern are technically still in the race for both the UEFA Champions League and Bundesliga.
However, it will take a monumental uptake in form for Bayern to truly consider themselves still in the race, despite their history and the quality they possess.
When Leipzig equalised it did not feel shocking or unexpected, because the Bavarians are so accustomed to conceding goals and dropping points.
It has got to the point now where you feel anything can happen when Bayern are involved, as the opposition may feel as though there’s a point or three to be had.
It is a vitally important win for the Rekordmeister, which showed a desire to fight to the end.
That is exactly what they need to keep doing between now and May if they are to push Leverkusen the whole way.
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Leipzig's inconsistency in 2024 continues
Leipzig showed at the beginning of the season in the DFL-Super Cup that they are capable of beating Bayern, winning 3-0 in the Allianz Arena.
On Saturday however, it was not a confident performance from Rose’s side. Despite an array of attacking talent in Xavi, Loïs Openda, Šeško and Dani Olmo, Leipzig were far from free-flowing.
Šeško in particular had several clear-cut chances, but failed to find the back of the net more than once. Leipzig actually had more shots on target, but lacked the clinical finishing shown by Kane for the opposition.
Bayern’s defence was fairly makeshift, as Eric Dier filled-in at right back and Alphonso Davies’ absence meant Raphaël Guerreiro was called upon.
Rose spoke highly of Bayern ahead of the game, but may have been guilty of showing them too much respect.
Despite the threat Bayern pose, there was a sense that this game may be clear opportunity to get a result on the back of the opposition’s negative momentum.
Leipzig’s form since the turn of the year will be concerning for Rose, as a difficult run of games approaches. The club is not firing on all cylinders.
Combined with Dortmund’s relative uptake in form and Stuttgart’s remarkably consistent season, adds further concern that they could miss out on a top-four position.
After the visit to Bochum, Leipzig take-on Köln and then SV Darmstadt.
Both of which are routinely causing problems for opposition far higher in the table, and desperate for points as crunch-time in the Bundesliga season approaches.