Two-time European champions Benfica, who lost to PSV in the 1988 final, are looking to return to club football's top table after losing in qualifying last season.
It was the fifth match of the campaign already for PSV, who have already knocked out Galatasaray and Midtjylland.
Portugal forward Rafa Silva gave the hosts the lead in the 10th minute, latching onto close-season signing Roman Yaremchuk's clever pass and sliding a low finish into the bottom corner.
Benfica grabbed their second goal shortly before half-time as German midfielder Julian Weigl turned home a loose ball in the box.
But PSV winger Gakpo dragged his side back into the tie less than six minutes after the break with a wonderful individual goal, embarking on a driving run from inside his own half before finding the net from 20 yards out.
Mario Goetze was denied an equaliser by an excellent tackle by Rafa Silva as PSV pushed to no avail.
Benfica know they can book a spot in the group stage draw, to be held in Istanbul on Thursday, August 26th, if they avoid defeat in Eindhoven next Tuesday.
August 18, 2021
Elsewhere, Swiss champions Young Boys took a big step towards only a second group stage appearance since the competition was revamped from the European Cup with a 3-2 win over Ferencvaros in Bern.
After early goals for both sides, Young Boys played for over an hour with 10 men after Silvan Hefti was sent off.
Myrto Uzuni missed the resulting spot-kick for the Hungarians and Vincent Sierro's stunning long-range strike put the home side in front in the 40th minute.
Ulisses Garcia's second-half effort put Young Boys in total control but Franck Boli's second goal of the game, with eight minutes remaining, gave Ferencvaros hope ahead of the return leg in Budapest.
The day's other match saw Malmo beat Ludogorets Razgrad 2-0 thanks to goals from Veljko Birmancevic and Jo Inge Berget in Sweden.
On Tuesday, Pedrinho scored the winner as Shakhtar Donetsk claimed a 1-0 victory in Monaco, while Salzburg got the better of Brondby 2-1 and Moldovans Sheriff Tiraspol brushed aside Dinamo Zagreb 3-0.
August 18, 2021