The home side's miserable record against BVB continued as the Westfalen men earned a resounding Bundesliga victory.
Christian Eriksen and Victor Wanyama were on target, but an own goal from Harry Kane and a red card for Dele Alli cost Mauricio Pochettino's side dear.
The Serie A side will take a two goal cushion into the second leg as their hosts played most of the game with 10 men.
Antoine Griezmann became the club's all-time top Champions League scorer in a priceless victory to give the Spaniards the advantage.
The Catalan side left it late to secure the points over the La Liga strugglers, though the pressure remains on the manager from the fans.
Jose Mourinho introduced both Paul Pogba and Zlatan Ibrahimovic in the second half and the pair combined to hit the winning goal late in the game.
The Welshman made a triumphant return after three months on the sidelines to sign off a routine victory for the hosts on Saturday.
The European champions got off to a bad start at the Santiago Bernabeu, but responded well with goals from Karim Benzema, Toni Kroos and Casemiro.