Sane's goal sealed a 1-0 victory over strugglers Arminia Bielefeld, who kept Bayern at bay for the opening 70 minutes at the Allianz Arena.
It was another surprisingly difficult domestic outing for Julian Nagelsmann's men after last week's shock home loss to Augsburg.
But Sane netted for the ninth time this season as Bayern, chasing a 10th straight Bundesliga title, restored their one-point advantage over Dortmund.
Bielefeld remain second-bottom in the table, four points from safety.
Dortmund had earlier gone top with a 3-1 win at Wolfsburg when Erling Haaland scored on his return from injury.
The return of Haaland is a huge boost for Dortmund, who host Bayern next weekend in a top-of-the-table clash.
Haaland netted his 50th Bundesliga goal in as many games, just eight minutes after coming off the bench in the second-half following a five-week absence with a leg injury.
He tapped home a Julian Brandt cross, adding to an Emre Can penalty and Donyell Malen goal for Dortmund after Wout Weghorst had given Wolfsburg an early lead.
"We missed him (Haaland) in the last few weeks. He's a weapon," Brandt told Sky.
The win came three days after a 3-1 defeat at Sporting Lisbon ended Dortmund's hopes of reaching the Champions League last 16.
"We needed to show a reaction after the game on Wednesday," said Can, who was sent off in Lisbon.
Third-placed Freiburg are now eight points adrift of second-placed Dortmund after their 2-1 defeat at Bochum handed the visitors a third straight league loss.
Serbian striker Milos Pantovic scored the winner eight minutes from time to ruin Christian Streich's 300th Bundesliga game as Freiburg coach.
"It would have been a cool day if we had won," admitted Streich.
Bayer Leverkusen can leapfrog Freiburg into third if they win at Covid-hit RB Leipzig on Sunday.
Leipzig have Willi Orban, Yussuf Poulsen, Hugo Novoa and Mohamed Simakan all in quarantine after testing positive.
However, goalkeeper Peter Gulacsi could return if a PCR test is negative Sunday.
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