With the Bundesliga champions 1-0 up heading into the second half at Signal Iduna Park, the 30-year-old was given the perfect chance to double the visitors' lead.
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Leon Goretzka, who fired Julian Nagelsmann's side into the lead in the 33rd minute, latched onto a through ball from Joshua Kimmich as BVB goalkeeper Alexander Meyer raced out.
But the shot-stopper failed to claim the lofted pass as Goretzka beat him to the ball where he smartly cushioned the ball back to a waiting Mane ten yards out from goal.
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However the former Liverpool winger got his header in the 49th minute all wrong as he put it wide of the post.
The Senegalese star was left shocked after he failed to hit the target with no Dortmund player in his way.
And it proved to be costly as Dortmund came back from two goals down to nick a point at the death.
It looked as though Leroy Sane had spared Mane's blushes when he put Bayern 2-0 to the good four minutes later.
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Just like Goretzka's opener, Sane took aim from distance as his shot rifled into the bottom left corner.
The ex-Manchester City man is enjoying a fantastic season so far having chipped in with nine goals and three assists in 14 matches.
He is Bayern's top scorer this campaign with the 26-year-old enjoying arguably his best football since returning to Germany in 2020.
Dortmund then pulled a goal back in the 74th minute through teenage starlet Youssoufa Moukoko.
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The 17-year-old smashed the ball home beyond the outstretched arms of Manuel Neuer with a fiercely-struck shot from the middle of the box.
In doing so, he became the youngest-ever goalscorer in a Bundesliga fixture between the two teams.
Dortmund pressed for a late equaliser and they got their reward in the 95th minute after Anthony Modeste headed home from six yards out.
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The former Blackburn striker got on the end of Nico Schlotterbeck's whipped in cross in what turned out to be the last kick of the game.
A win for Bayern would have taken them back to the top of the table ahead of Union Berlin, but they remain in second spot.
As for Dortmund, they are up to fourth - level on points with Bayern but behind on goal difference.