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The German international used his right foot to expertly bring down a long ball before hammering home with his left foot to give Dortmund a 36th-minute lead.
After a poor first half, PSG missed several chances to equalise in the second, with Kylian Mbappe curling an effort off the inside of the post before Achraf Hakimi struck the woodwork moments later.
Fullkrug should have scored a second after great work down the right flank from on-loan Manchester United winger Jadon Sancho but the striker could only blaze over the bar from five yards out.
In their third Champions League meeting of the season, Dortmund got the better of the Ligue 1 winners with the directness 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 opportunities.
Marcel Sabitzer had a super chance to double Dortmund’s lead on the stroke of half-time but his close-range volley was superbly saved by Gianluigi Donnarumma.
PSG manager Luis Enrique rallied his side at the break and Mbappe’s first major moment in the game came in the 51st minute with the France captain hitting the post with a trademark cut in from the left.
Moments later the ball fell to former Dortmund right-back Hakimi and his effort, just like Mbappe’s, stuck the inside of the opposite post and ran to safety.
Ousmane Dembele, another ex-Dortmund player, blazed a glorious chance over the bar with 10 minutes left while Fullkrug though he deserved a penalty after being nudged in the back when he appeared set to head in Julian Brandt’s cross.
The 1-0 scoreline leaves this tie finely balanced for the second leg at Parc des Princes on Tuesday.
England international Sancho was the best player on the pitch at the Signal Iduna Park.
PSG left-back Nuno Mendes did not know how to deal with the lightning quick feet of the Manchester United loanee and now has six days to find a way to deal with the 24-year-old before the second leg.
In one brilliant second-half moment, Sancho feigned to go left before dinking the ball over the leg of Mendes and then 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.