Dayot Upamecano and Konrad Laimer scored for the defending champions, but Serhou Guirassy and a late strike from Fabio Silva ensured Dortmund left Munich with a valuable point.
The result kept both sides level on points at the top of the Bundesliga table, with Bayern maintaining their advantage on goal difference.
Laimer gives Bayern first-half advantage
Bayern started strongly and nearly took the lead in the 10th minute when Michael Olise struck the crossbar with a curling effort.
Dortmund thought they had broken the deadlock in the 34th minute through Jobe Bellingham, who finished after a defensive mistake. However, the goal was disallowed following a VAR review, which identified a foul by Carney Chukwuemeka on Bayern goalkeeper Jonas Urbig.
The hosts eventually took the lead in the 38th minute. Olise capitalised on a misplaced pass from Felix Nmecha before playing Laimer through on goal. Gregor Kobel saved the Austrian's initial effort, but Laimer reacted quickly to convert the rebound.
Bayern went into the interval with a deserved 1-0 advantage.
Kane misses penalty as Dortmund fight back
Bayern had an opportunity to extend their lead shortly after the restart when Harry Kane was awarded a penalty in the 56th minute following a handball by Nmecha.
However, the England captain struck the crossbar with his spot-kick, allowing Dortmund to remain in the contest.
The visitors punished Bayern four minutes later. Nico Schlotterbeck initiated the move before Chukwuemeka squared the ball for Guirassy, who finished clinically to make it 1-1 in the 60th minute.
Bayern responded almost immediately. Joshua Kimmich delivered an excellent cross in the 66th minute, and Kane headed the ball across goal for Upamecano to finish from close range.
The goal restored Bayern's advantage at 2-1.
Fabio Silva strikes late to rescue Dortmund
Bayern appeared to be heading towards another important victory, but Dortmund refused to surrender.
The visitors continued searching for an equaliser and were rewarded in the 89th minute. Konstantinos Karetsas charged down Aleksandar Pavlović's attempted clearance, allowing Fabio Silva to collect the loose ball and fire a low finish into the far corner.
The late goal brought Dortmund level for the second time and silenced the home supporters.
Bayern nearly snatched a winner in stoppage time when substitute Lennart Karl struck the crossbar, but Dortmund held on to secure a hard-fought draw.
Bayern maintain unbeaten record
The draw will feel like a missed opportunity for Bayern, who created several promising chances but failed to convert their dominance into three points.
Kane's missed penalty proved particularly costly, while Dortmund deserve credit for their resilience and determination to fight back on two occasions.
Konrad Laimer was named Man of the Match after opening the scoring and delivering an influential performance in midfield.
The result leaves Bayern and Dortmund level on points at the summit of the Bundesliga, keeping the title race finely balanced after eight league matches.
Full-time: Bayern Munich 2-2 Borussia Dortmund
Goals:
• Bayern Munich: Konrad Laimer 38', Dayot Upamecano 66'
• Borussia Dortmund: Serhou Guirassy 60', Fabio Silva 89'
Key moment: Harry Kane missed a second-half penalty, striking the crossbar in the 56th minute.