Kane brace helps Bayern keep pace with leaders Leverkusen
Leverkusen's win meant the perennial German champions started seven points behind but Kane settled nerves by netting after two minutes, tapping home a pass from Leroy Sane.
Bayern lost midfielders Leon Goretzka and Joshua Kimmich to flu just hours before the match and had only five outfield players on the bench but dominated a listless Stuttgart.
"It was pretty intense today," Bayern coach Thomas Tuchel said of his side's injury crisis, revealing "I tried to stay very calm and keep my distance on the sidelines to give the players a bit more trust."
"Today was very good, but it's not over, we've got another game."
Bayern had two more goals struck off for offside in the opening half but Kane doubled up early in the second, heading in from close range from a free-kick.
Kim Min-jae was next to score, heading in from another set piece for his first goal in Bayern colours, sealing the game for the home side.
Kane told DAZN that it was "probably our best performance of the season", saying "everyone who came in did really well. We'll need that throughout the season".
Bayern sit four points behind Leverkusen and have a game in hand, against 15th-placed Union Berlin, in January.
Kane's double took him to 20 league goals from 14 matches so far this season -- four clear of Stuttgart's Serhou Guirassy, who was kept quiet by Bayern's miserly defence.
Brilliant Boniface
Earlier on Sunday, Nigeria striker Victor Boniface scored and laid on two assists as Leverkusen beat Frankfurt 3-0, continuing his side's hopes of a maiden Bundesliga title.
Before the match, the Leverkusen fans unveiled a huge Christmas wishlist, calling on the club to make their Bundesliga title dreams come true.
"We've won nothing yet. After half a year nobody wins anything, especially not a league title," Leverkusen midfielder Granit Xhaka said to DAZN.
"We've got another game midweek and then we go on holiday, where we need to be careful what we do. Then we'll come back and step on the gas ahead of the rest of the season."
Xabi Alonso's unbeaten Leverkusen have won 21 and drawn three of 24 matches in all competitions this season.
Boniface squeezed a low shot under Kevin Trapp to open the scoring 14 minutes in.
The Nigerian then created Jeremie Frimpong's goal early in the second half, his shot rebounding off Trapp and landing perfectly for the Dutch international to slam home.
The 22-year-old, who is almost certain to be selected for Nigeria's Africa Cup of Nations bid in January, then helped seal the result, finding Florian Wirtz in space for Leverkusen's third.
Wirtz, who was again superb while pulling the strings in midfield, hit the crossbar with 10 remaining.
Frankfurt have won just one of their past eight matches in all competitions -- last Saturday's surprising 5-1 home thrashing of Bayern.
Freiburg's Michael Gregoritsch and Roland Sallai came off the bench to secure a 2-0 home win over 10-man Cologne.
Dogged but outclassed, Cologne fought hard but were reduced to 10 men with just over half an hour remaining when Jeff Chabot swept Lucas Hoeler's legs out from under him, picking up a second yellow.
Hoeler was replaced by Gregoritsch and the 1.93-metre tall Austrian had Freiburg ahead shortly after.
Gregoritsch was first to a loose ball in the Cologne box, slamming home to set his side on course for their seventh win of the season.
Gregoritsch has now scored in his past league three games, having broken a goal drought which dated back to April.
"I just keep scoring," a thrilled Gregoritsch told DAZN.
"I've been in the right place at the right time... of course my self-confidence is coming back a little."
Sallai, who was also brought on with Gregoritsch, scored in the fifth minute of added time to seal the result.