The spot-kick was awarded after Christian Eriksen was found guilty of handballing in the box as VAR intervened to give the penalty.
And with Bayern needing a goal to restore their two-goal lead, Kane made no mistake as he delivered a powerful effort into the corner of Andre Onana's goal.
It came at a crucial time for the German giants, who saw a resurgent United come out after half-time.
After a dismal half that saw the Red Devils down by two, Rasmus Hojlund scored his first goal for the club since signing in the summer.
A neat exchange with Marcus Rashford saw the Denmark international hit low before the ball deflected off a defender and past Sven Ulreich.
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It halved the deficit and there was a new-found belief for the visitors at the Allianz Arena.
But that was snuffed out minutes later when Eriksen was adjudged to have handballed in the area.
While the decision might be seen as harsh by English viewers, UEFA's handball rules are far more lax than in the Premier League.
Nevertheless, Kane was not going to give up the chance to score on his Champions League debut - especially against United.
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The former Tottenham star made no mistake from the spot, putting Bayern back in control of the contest.
Kane has been in fine form since moving to the Bundesliga this year.
In four appearances, he has already netted four in the league as he looks to end his trophy hoodoo.
Bayern are the clear favourites to win the league but the Champions League remain their main ambition.
The Bavarian outfit have not won the competition since 2020 but remain one of the best clubs to play in the tournament.
Only two clubs have more Champions League trophies than them in Real Madrid and AC Milan while their total of six is matched only by Liverpool.