Niclas Fullkrug a German international hammer home with his left foot to give Dortmund a 36th-minute lead.
After a poor first half performance, PSG missed several chances to equalise in the second, with Mbappe curling an effort off the inside of the post before Achraf Hakimi struck the woodwork moments later.
After great work down the right flank from on-loan Manchester united winger Jadon Sancho, Niclas Fullkrug should have scored a second but the striker could only blaze over the bar from five yards out.
Dortmund got the better of the Ligue 1 winners in their third Champions League meeting of the season with the direction of their play, their aggression and tenacity.
PSG controlled much of the first half but failed to record a shot on target while Dortmund carved out better opportunities.
PSG manager Luis Enrique rallied his side at the break and Mbappe’s first major moments in the game came in the 51st minutes with the France captain hitting the post with a trade mark cut in from the left.
Moments later the ball fell to former Dortmund right-back Hakimi and his effort, just like Kylian Mbappe’s stuck the inside of the opposite post and ran to safety.
Another Dortmund ex-player, Ousmane Dembele, blazed a glorious chance over the bar with a 10 minutes left as Fullkrug though he deserved a penalty after being nudged in the back when he appeared set to head i Julian Brandt’s cross.
Meanwhile, England international Sancho was the best player on the pitch at the signal Iduna Park.
In one brilliant second-half moment, Sancho feigned to go left before dinking the ball over the leg of Mendes and the delivering a pinpoint cross which Fullkrug hit over the bar.
Sancho has had plenty of doubters after a difficult spell at Old Trafford following his £73m move from Dortmund in 2021.
The 1-0 score line leaves this tie finely balanced for the second leg at parc des princess on Tuesday.
