Only four minutes had elapsed when visiting goalkeeper Matej Kovar had his palms stung for the first time, diving low to his right to deny Ridle Baku.
Ruben van Bommel then had a somewhat tamer strike saved at the other end, before the Dutch side profited from a controversial penalty awarded for handball against Baku after VAR intervention.
Ivan Perisic took responsibility, but his low drive down the middle was expertly kept out by the trailing leg of Maarten Vandevoordt.
It had been a lively opening 20 minutes and the chances continued to flow for both teams. Ezechiel Banzuzi and Antonio Nusa missed the target for Martin Demichelis’ men, either side of Armando Obispo nodding wide for PSV.
Gruda’s driving run into the area then resulted in another strong stop by Kovar, while Romulo Cardoso - twisting and turning to make space for himself 20 yards out - blazed harmlessly over the bar as Leipzig headed to half-time with a match poised at 0-0 for the first time since mid-April.
The second period was just as intriguing as the first, with Nusa and Baku forcing sharp saves out of the goalkeeper with low drives towards the near post.
Those chances came either side of Sven Mijnans haplessly spraying wide from the penalty spot at the other end, while Vandevoordt palmed away Filip Kostic’s thunderbolt.
The Leipzig stopper also thwarted Sergino Dest, who profited from a clever Perisic reverse pass, before his opposite number made a stunning stop from point-blank range from Gruda.
There had been 23 shots across the opening 60 minutes, 11 of them on target, but deadlock remained. Gruda would finally make the breakthrough with a stunning effort from outside the area, though, chesting the ball down and unleashing a left-footed shot that flicked off Perisic before flying into the back of the net.
The win is Leipzig’s second across their last 12 UCL fixtures, but it did not come without a hiccup. Only a long VAR check ruled out Obispo’s strike for offside as PSV failed to score in a game for the first time since their opening match of the campaign and remain winless in the league phase (D1, L1).
Flashscore Man of the Match: Brajan Gruda (RB Leipzig)