A late header thwarted the Euro 2016 semi-finalists, who remain undefeated but find themselves third in their pool.
The reigning champions fell behind to a superb early strike from Kerem Demirbay but managed to pick up a point with the visitors remaining just four points behind in the table.
The Manchester United midfielder caught out Maarten Stekelenburg as he scored from distance in the first half, with Didier Deschamps's side holding on to claim all three points.
The World Cup holders were far too strong on Saturday, and strolled to victory thanks to the Bayern star's efforts while Toni Kroos also found the net.
The two clubs will suffer the same fate as Barcelona in 2014 after breaches of rules around the protection of minors, meaning they will not be allowed to sign players until 2018.
The Barcelona wizard wasted no time in announcing his return to international football after ending his short-lived retirement, breaking the Celeste open in the first half.
Two goals from Karim Benzema had put the hosts in a commanding position, but they failed to kill off the game and were made to pay by the Bundesliga club.
The Spaniard was in fine form in front of goal as Arsene Wenger's men pipped Paris Saint-Germain to first place in Group A with a dominant victory on the road on Tuesday.