Dortmund dominated the game for large periods but saw their celebrated centre-forward fluff a second-half penalty amid a series of modest finishes and an outstanding goalkeeping performance from Ederson.
Greece striker Mitroglou succeeded where Aubameyang had failed, converting scrappily but decisively from a 48th-minute corner to put the underdogs in charge heading into the second leg.
Both sides were short of key men with Benfica’s Brazilian striker Jonas and Lisandro Lopez both injured and Andrija Zivkovic suspended. Dortmund, meanwhile, saw Mario Gotze fail a fitness test to join Sven Bender, Nuri Sahin and Sebastian Rode on the treatment table.
Benfica were quick to show their attacking intent, Eduardo Salvio making headway down the right in the opening seconds and Nelson Semedo earning an early corner.
The sides exchanged half-chances, Ousmane Dembele seeing his first chance of the game well closed down and Salvio skying over after weak defending from Sokratis Papastathopoulos.
In the 11th minute the visitors produced something much more clearcut but Aubameyang was let down by his finish. Dembele did well to steal possession for his side and quickly located his partner’s run on goal.
The Gabon striker was perfectly positioned to punish Benfica but rushed his effort and lifted it high over the crossbar.
Dembele was becoming the game’s dominant influence and came close after 23 minutes, profiting from the good work of Marco Reus and Lukasz Piszczek. His shot was goalbound but goalkeeper Ederson, assisted by Victor Lindelof, protected their line.
The pair needed attention from the physio after colliding but Ederson was back up and running in time to get a fist on the resulting corner.
Dortmund kept coming, Raphael Guerriero not allowing Ljubomir Fejsa to usher the ball out of play and touching the ball across goal with Ederson beaten. Aubameyang was sniffing around but could not get the tiny touch needed to find the open net.
It was goalless at the break, Dortmund nursing a minor grievance before the whistle when Ederson’s foul on Dembele, just outside the box, went unpunished.
Benfica came out at a much improved tempo and grabbed the lead just three minutes into the second half.
After forcing a corner Luisao headed Luis Pizzi’s cross towards Mitroglou, who needed a couple of attempts to beat Roman Burki from close range.
It still looked a fragile lead and Aubameyang had yet another yawning chance to net five minutes later. Once again, though, his finish was lamentable.
Reus and Marcel Schmelzer tested Ederson before Fejsa’s handball gave Aubameyang a shot at redemption from 12 yards. But his spot-kick was no better than his previous attempts, a soft strike straight down the middle where the waiting keeper had no trouble.
Thomas Tuchel substituted the misfiring hitman moments later but there was no cutting edge to be found elsewhere either. Christian Pulisic came closer than most after stepping off the bench, drawing Ederson’s best save yet in the 84th minute.
His powerful shot took an awkward deflection but the Brazilian stopper refused to let his clean sheet slip, ensuring victory against the odds and the run of play.