Vinicius Junior gave Real Madrid a narrow half-time lead at the Allianz Arena, before Leroy Sane fired in at the near post to pull Bayern level in the 53rd minute.
Just four minutes later, Harry Kane kept his composure to dispatch his spot-kick after Jamal Musiala was brought down in the box.
Bayern's hopes of claiming a first-leg win were quashed when Kim Min-jae gave away a penalty to present Vinicius with the chance to complete his brace.
Kane makes Champions League history
Although his side was unable to protect their slender advantage, Kane was still able to celebrate a successful evening on a personal level.
The forward set a new record for the most goal involvements by an English player in a single Champions League campaign after scoring eight goals and providing three assists in his 11 European appearances this season.
Having registered 11 goal involvements, Kane has surpassed the record he previously shared with David Beckham and Steven Gerrard.
Beckham achieved 10 goal contributions for Manchester United in 1998-99, while Gerrard matched that figure for Liverpool in 2007-08.
Full focus on next week's second leg
As a result of his penalty against Real Madrid, Kane has now scored 43 goals in as many matches since joining Bayern from Tottenham Hotspur last summer.
Although he is pleased with his prolific campaign, the England captain's main focus is to help Bayern progress to the Champions League final at Wembley.
Kane told TNT Sports: "It's been a good season so far but, like I always say, it's about the next one, the next game and, of course, the big game next week away from home against Real Madrid.
"Everything we're fighting for this year is in this competition so we've just got to find a way to get it done. It's going to be tough, obviously, the Bernabeu, Real Madrid in the Champions League but we just got to go there with full belief."
Thomas Tuchel's side will face Stuttgart in Saturday's Bundesliga clash, before they travel to the Spanish capital for next Wednesday's second leg.
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