Trailing 3-1 from the first leg, the visitors knew they needed to make a fast start, and came close to reducing their deficit through an early Miha Zajc chance that was deflected narrowly wide.
Mounsef Bakrar then saw a fizzing shot from an awkward angle brush the wrong side of the post, while Bruno Goda thundered an effort against the crossbar from outside the box.
In between those two chances, Daan Heymans saw his flicked header superbly turned around the post at the other end, before Yira Sor blazed over from a central position.
That wayward finishing proved costly when the Blues took the lead on the night, although they had opposition goalkeeper Tobias Lawal to thank as he gifted possession to Bakrar inside the area with a horrific pass.
The forward could barely believe his luck and duly tucked home on the stroke of half-time to offer the visitors extra hope heading into the second half.
Nicky Hayen must have given his side a good talking to during the interval, though, as they were much improved in the second half. Their equaliser came within six minutes as Sor atoned for his earlier miss by brilliantly whipping his shot into the back of the net after being teed up by Bryan Heynen.
Moments later, Heymans curled just the wrong side of the post from 20 yards, before Dinamo received another lifeline as Junya Ito tripped Gabriel Vidovic. The referee immediately pointed to the spot, and Luka Stojkovic found the bottom corner with the ensuing penalty.
Robin Mirisola could have pulled the hosts level shortly afterwards with a low strike, although Dominik Livakovic blocked it with his legs before also keeping Sor at bay.
Those saves proved crucial as Mario Kovacevic’s men struck gold by levelling the tie at 4-4 on aggregate as Stojkovic found the top corner with an outstanding 25-yard effort that flew into the top corner.
Genk were stunned, but this was familiar territory for a side who had not won a major European knockout game on home soil since 2017. Things nearly got worse in injury time too, but a tame Cardoso Varela shot was saved to ensure extra time was required to decide the tie, the first ten minutes of which was a non-event.
The Belgian outfit eventually went back in front in the tie, though, when Ito brilliantly timed his run onto Nikolas Sattlberger’s pass and poked the ball in with the help of a deflection off Matteo Pérez Vinlöf.
That sucked the life out of Dinamo, who received another blow when VAR intervened to give a straight red card to Stojković for slapping Matte Smets in the face. Genk looked to make the most of their man advantage, only for Aaron Bibout to have a stab goalwards saved.
They finally put the tie to bed with six minutes left, when Jarne Steuckers drove forward and fed Heymans, who duly converted his fourth UEL goal of the season to make it 3-3 on the night.
Flashscore Man of the Match: Daan Heymans (Genk)