Severin Nioule's goal just after the hour mark in Slovakia last week had been the difference maker in the first leg, and Hacken looked set for a more comfortable night 13 minutes in, when the visitors' Patrick Nwadike committed a foul in the penalty box.
Simon Gustafson made no mistake from 12 yards to give the Svenska Cupen winners the lead on the night and a 2-0 aggregate advantage.
But with their backs to the wall, Trnava began to create opportunities - Michal Tomic, Cedric Badolo and Martin Mikovic all came close before the break, and Etrit Berisha in the Hacken goal was preserving the lead.
20 minutes after the break, the goal the Slovakian side deserved finally arrived, and this time there was nothing Berisha could do as Roman Prochazka's free-kick flew past him to level the second leg at 1-1.
Eight minutes from time, they were level in the tie - Julius Lindberg had come within a whisker of settling the tie, only for Trnava to go down the other end and send Giorgi Moistsrapishvili through, and he made no mistake in firing past Berisha.
Few late opportunities meant the two sides were preparing themselves for an extra 30 minutes, but Dembe had other ideas. In the 96th minute, Gustafson's corner caused havoc in the Trnava box, and Dembe pounced to level the second leg and send Hacken through, 3-2 on aggregate.
There was still time for Silas Andersen to receive a second yellow card, but Hacken had got the job done and will move on to the Second Qualifying Round, where they will face Belgian giants Anderlecht.
