Ferencvaros hosted FC Twente after grabbing a 2-1 lead from the first leg, courtesy of Lenny Joseph's brace in Enschede.
A slow first 20 minutes were broken open by Thomas van den Belt, who fouled Kristoffer Zachariassen after a quick Hungarian combination. Belgian Toon Raemaekers stepped up and calmly slotted the ball in the bottom left corner, giving Ferencvaros a virtual two-goal lead on aggregate.
FC Twente were handed a boost when former Eredivisie player Philippe Rommens was handed a red card following a harsh tackle on Ramiz Zerrouki, leaving Ferencvaros shorthanded with an hour left to play.
Even against ten men, FC Twente hardly showed up dangerously in Ferencvaros' half and even saw their Hungarian opponents double their lead through Lenny Joseph, who ruthlessly punished the visitors on the counter and left Lars Unnerstall rooted to the ground by burying the ball past the German's near post.
Twente attempted to up the ante and moved up more the later it got, but Sam Lammers and Daouda Weidmann couldn't get past Denes Dibusz as time ran out for the Tukkers.
With ten minutes to go, Wout Weghorst pulled Twente back within one by extending Marko Pjaca's header behind Dibusz. The visitors were given a definite lifeline when Lucas Vennegoor of Hesselink, son of former Hull City and Celtic striker Jan Vennegoor of Hesselink, headed in Twente's second goal shortly after, making the final 10 minutes crucial.
Countless crosses were sent into Ferencvaros' box, but many lacked real quality as the clock ran out and FC Twente were officially demoted to the UEFA Conference League qualifiers.
Ferencvaros survived in their own Groupama Arena and progressed to the third round of the UEFA Europa League qualifiers, in which they'll face Fenerbahce. FC Twente will face FC DAC 1904 in the UEFA Conference League.