Leipzig's defeat on Saturday means it can no longer overhaul Bayern, which holds a seven-point lead before hosting Borussia Monchengladbach in the late game. Two rounds remain after this weekend's games.
Jadon Sancho scored twice for Dortmund after captain Marco Reus fired his team into a seventh-minute lead. Sancho capitalized on some lackluster defending to make it 2-0 in the 51st.
Germany defender Lukas Klostermann pulled one back for the visitors in the 63rd, and Leipzig's equalizer came in the 77th when Hwang Hee-chan shrugged off Mats Hummels' challenge to set up Dani Olmo.
But Sancho saved the best for last, playing Raphaël Guerreiro in with his heel before running to accept the ball back and score in the 87th.
It's Bayern 30th Bundesliga title and its 31st German championship including the 1932 title. The Bundesliga was formed in 1963 and the rest of Bayern's titles came after its promotion in 1965.
Dortmund moved into fourth - the last qualification spot for the Champions League - though Eintracht Frankfurt can reclaim the position if it beats Mainz in their derby on Sunday.
Dortmund and Leipzig will meet again in Berlin on Thursday for the German Cup final.
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