Bayern were on course to start thinking of title celebrations as substitutes Serge Gnabry and Raphael Guerreiro both scored within four minutes of each other to cancel out Maximilian Beier's opener for Dortmund after the break on Saturday.
But Dortmund defender Waldemar Anton equalised on a rebound in the 75th after Jonas Urbig saved Serhou Guirassy's acrobatic effort.
It stopped Bayern from capitalising on defending champions Bayer Leverkusen's 0-0 draw with Union Berlin earlier, as the Bavarian powerhouse stayed six points clear at the top with five rounds remaining.
It's just a little bump for Bayern, who next need to come from a goal behind at Inter Milan in the second leg of their Champions League quarter-final on Wednesday. The final is being held at Munich's ground in May.
Leverkusen star Florian Wirtz returned after five weeks out but could not inspire his team to pick up anything more than a scoreless draw against Union.
Wirtz couldn't return soon enough for Leverkusen, who were knocked out of the Champions League by Bayern and the German Cup by third-division Arminia Bielefeld while the 21-year-old recovered from an ankle injury.
Leverkusen coach Xabi Alonso sent him on in the 57th minute and he made an immediate impact, setting up a chance for Exequiel Palacios.
Union forward David Preu was booked on his Bundesliga debut for a bad challenge on Wirtz's ankle in the 83rd but the striker battled on and remained central to all of the home team's attacking play.
Union goalkeeper Frederik Ronnow was alert to a Wirtz cross before Granit Xhaka fired Leverkusen's last chance over from a free kick.
"We've dropped too many points at home. I don't know why," said Leverkusen goalkeeper Lukas Hradecky, whose side completed an unprecedented unbeaten league and cup double last season.
"As of now, we're a point closer to Bayern but of course we'd hoped it would be three."
Union's fifth straight game without defeat handed the initiative to Bayern.
"It's difficult now, but the way I know Leverkusen, they'll finish the season well. You never know in football," Union captain Christopher Trimmel said of Leverkusen's title chances.
Also on Saturday, Hoffenheim defeated Mainz 2-0 to leave Leipzig in fourth place, while Freiburg came from behind to beat Borussia Monchengladbach 2-1, Augsburg won at Bochum 2-1 and Jackson Irvine's St. Pauli won a relegation battle at bottom side Holstein Kiel 2-1.
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