Hacken had given themselves a lot to do away in Norway, having lost the home leg 2-0, one of three games without a win in all competitions since knocking Anderlecht out in the previous round.
Yet the tie could have been all but over within minutes had Hacken goalkeeper Etrit Berisha not denied Felix Myhre and then Saevar Magnusson in quick succession.
Then before 15 minutes had been played, Berisha was called upon again with a brilliant save to his right after his defence afforded Eggert Gudmundsson plenty of time and space inside the box.
Having soaked up the pressure, Hacken grew into the game and began creating chances in the final 15 minutes of the half, with Mikkel Rygaard and Julius Lindberg testing Matias Dyngeland in the Brann goal.
Unlike their Norwegian opponents, the Swedish side turned that pressure into a goal, getting themselves back into the tie with seven minutes of the first half to play, and in fortunate circumstances as Denzel De Roeve turned the ball into his own net.
Despite setting themselves up for a crucial second half, Hacken's comeback failed to materialise. In fact, the Svenska Cupen winners did not produce a shot on goal after the break, with the best chance arguably falling to Adrian Svanback, whose header sailed just over.
In added time, Brann had the chance to kill the tie off once and for all, when a melee in the Hacken box resulted in VAR intervening and awarding a penalty to the hosts. Niklas Castro to the responsibility, but Berisha's great night continued as he guessed right and made the save.
For Berisha and his teammates, it wasn't enough to stop Brann progressing to the UEL Playoffs to face AEK Larnaca or Legia Warsaw. Hacken will drop into the Playoff Round of the UECL, where they will take on the loser of the UEL tie between CFR Cluj and Braga.