Champions Bayer Leverkusen piled on the pressure on Friday with their own 3-1 win at home to Bochum, but Bayern avoided dropping points for the third straight game against fourth-bottom St Pauli.
Bayern, shaky before the international break, were without the injured Manuel Neuer, Alphonso Davies, Dayot Upamecano, Aleksandar Pavlovic and Kingsley Coman, while St Pauli had their Australian skipper Jackson Irvine back in the side after his midweek heroics in the World Cup win in China.
St Pauli hit the bar early on, but home nerves were settled by Harry Kane opening the scoring from Michael Olise's cross after the Hamburg-based visitors had lost the ball.
Elias Saad survived an offside video check to equalise past 21-year-old Bayern keeper Jonas Urbig on 27 minutes.
Kane thought he had put Bayern back ahead soon afterwards but this time the offside call stood.
Sane, a free agent in July at present after Bayern decided to prioritise other contract extensions, was given far too much room to bear in on goal and score on 53 minutes.
He netted his second from Kane's pass as Bayern steadied the ship with seven games to go, despite a cracking late St Pauli goal from Lars Ritzka.
"For me, it is only about trying to win games. We will all come together and find a solution (on the contract)," said Sane, after his ninth Bundesliga goal of the season.
Bayern also finished the game with 10 men after substitute defender Hiroki Ito limped off when they had used all their allotted changes, and there was fear the Japanese, who missed most of the season with a broken foot, had suffered an injury in the same spot.
"He felt something in the same foot where he had problems before," coach Vincent Kompany told reporters. "It is hard to say exactly what it is. He will have a scan and we'll see. It's not good for us."
Elsewhere, Eintracht Frankfurt moved closer to securing a Champions League spot after Mario Gotze clinched a 1-0 home win over 10-man VfB Stuttgart.
Frankfurt flew up to third, although Mainz would reclaim that spot with a win at 12th-placed Borussia Dortmund on Sunday.
Borussia Monchengladbach boosted their outside hopes of Champions League football next season by beating top-four rivals RB Leipzig 1-0 at home through Alassane Plea's header.
Gladbach move up to fifth while Leipzig, Champions League entrants the last six years, are down to sixth.
Heidenheim coach Frank Schmidt took charge of his 700th competitive game for the club and his side celebrated with a 1-0 win at Wolfsburg.
The victory lifted Heidenheim into the relegation play-off spot and dropped second-bottom Bochum into the automatic drop zone.
Hoffenheim are just two points above St Pauli and five ahead of Heidenheim after a 1-1 home draw with Augsburg, who scored early in the second half through Samuel Essende's first touch after coming on at the break. It was the quickest goal by a Bundesliga substitute since records began.
Andrej Kramaric levelled from the spot after a debatable handball.
Marvin Ducksch's fine free-kick from a tight angle helped mid-table Werder Bremen to a 3-0 win at rock-bottom Holstein Kiel.
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