Jude Bellingham and Mahmoud Dahoud scored late goals as Borussia Dortmund came from behind to seal a dramatic 3-2 win at Eintracht Frankfurt.
This was Bayern's seventh straight win over Dortmund, including five Bundesliga victories and two German Super Cup triumphs.
Sane's strike was Bayern's 102nd league goal of 2021, breaking Cologne's 44-year-old record for the most Bundesliga goals scored in a calendar year.
Freiburg remain the only unbeaten team in Germany's top flight and rose to third with a 2-0 win at Wolfsburg
First-half goals by Thomas Mueller and Joshua Kimmich put Bayern 2-0 up before Pavard was shown a straight red card just three minutes into the second-half.
Haaland made sure of the three points for Dortmund when he smashed his sixth goal this season into the net in the 91st minute.
Bundesliga: Serge Gnabry strikes twice as Bayern Munich overcome Cologne; Union Berlin draw with Hoffenheim
Gnabry gave Nagelsmann his first Bundesliga win as Bayern head coach by smashing the winner into the roof 19 minutes before the whistle.
Bundesliga: Erling Haaland nets twice, assists thrice as Dortmund rout Frankfurt; Stuttgart beat Greuther Fuerth
Defending champions Bayern Munich, who are bidding for a tenth straight league title, kicked off the new season with a 1-1 draw at Moenchengladbach on Friday.
Leipzig thought they had won the game at the death when Yussuf Poulsen headed in but he was penalised for handball after the referee consulted the pitchside monitor.
Leverkusen stayed sixth, nine points off the Champions League places. Hoffenheim remained 12th. Both teams have had a disappointing start to 2021.
Bundesliga: Bayer Leverkusen continue poor form with loss against Freiburg; Union Berlin held to 1-1 draw at Hoffenheim
Having topped the Bundesliga in December, Leverkusen are now five points adrift of the Champions League places in sixth after winning just one of their last six games.
Making his first appearance since his injury in the Berlin derby back in December, the former Germany international netted his seventh goal of the season in a hard-fought draw in Berlin.
Bundesliga: Erling Haaland spares Borussia Dortmund blushes in Hoffenheim draw; Bayer Leverkusen drop points
Erling Braut Haaland's 81st minute strike averted a fourth defeat in five Bundesliga games for crisis-hit Borussia Dortmund.
UEFA moved Manchester City's Champions League last-16 game at Borussia Monchengladbach to a neutral venue in Hungary
Bundesliga: Eintracht Frankfurt back to fourth spot with Hoffenheim win; Werder Bremen-Arminia Bielefeld match postponed
Filip Kostic steered Eintracht Frankfurt back to fourth in the Bundesliga with a 3-1 win at Hoffenheim on Sunday, as the day's late game was called off due to a heavy snowstorm sweeping across Germany.
Bundesliga: Bayern Munich stay seven points clear at top after win over Hoffenheim; Borussia Dortmund also claim victory
The European champions exacted perfect revenge for their only defeat of 2020 after losing by the same 4-1 scoreline at Hoffenheim last September.
What to watch in European football this weekend: AC Milan wavering; unfinished business for Lionel Messi
A quick look at key points in European football over the weekend across LaLiga, Serie A, Bundesliga and Dutch Eredivisie.
After title rivals RB Leipzig, Bayer Leverkusen and Borussia Dortmund all lost earlier in the weekend, three assists from Joshua Kimmich took Bayern Munich a step closer toward what would be their record-extending ninth consecutive Bundesliga title.
The victory was Leverkusen's first in five league games and puts them second on goal difference, level on 32 points with RB Leipzig, who host Union Berlin on Wednesday.
Bundesliga: Erling Haaland shines as Borussia Dortmund beat RB Leipzig; Schalke thrash Hoffenheim 4-0
Haaland is the first player to score 25 goals in his first 25 Bundesliga games and the 20-year-old has now netted 35 goals in 34 games in all competitions for Dortmund.