Juventus, Bayer Leverkusen, Roma and Seville reached the Europa League semi-finals, and the Conference League will feature West Ham, Basel, Fiorentina and AZ Alkmaar.
Roma were 1-0 down after the first leg in Netherlands, but Leonardo Spinazzola levelled the tie on the hour mark.
Igor Paixao then scored with 10 minutes of normal time remaining, leaving the away side on the cusp of progressing.
However the talismanic Paulo Dybala forced the game into extra time with a late goal, before Stephan El Shaarawy and Lorenzo Pellegrini scored in the additional half-hour to complete the comeback.
Juventus scored first through Adrien Rabiot before Marcus Edwards levelled from the spot on 20 minutes for hosts Sporting Lisbon, but the Italians held onto their advantage from the first leg to win 2-1 overall.
Bayer Leverkusen overwhelmed Union Saint-Gilloise in Belgium as Moussa Diaby, Mitchel Bakker, Jeremie Frimpong and Adam Hlozek scored for the away team, while Union managed a consolation goal from Casper Terho when his side were already 3-0 down.
The sides were level at 1-1 after the first leg but the Germans progressed comfortably to the final four.
Manchester United crashed out after a string of defensive errors from Harry Maguire and David de Gea saw Sevilla win 3-0 at home after a 2-2 draw in the first leg.
Youssef En-Nesyri scored twice and Loic Bade added one as the Spaniards sent Erik ten Hag’s injury-hit side out of Europe.
Seville will play Juventus in their semi-finals, with Roma to face off with Leverkusen.
In the Europa Conference League, Fiorentina reached the semi-finals when they hosted Polish side Lech Poznan, with the Serie A outfit starting 4-1 ahead on aggregate after the first leg.
Poznan cut the deficit with three goals from Afonso Sousa, Kristoffer Velde and Artur Sobiech leaving the scores tied, but Riccardo Sottil and then Gaetano Castrovilli scored late to clinch a 2-3 defeat for Fiorentina, and a 6-4 win overall.
Dutch side AZ Alkmaar are through after they levelled the aggregate scores against Anderlecht with a 2-0 win on the night with two earlier contributions from Vangelis Pavlidis, and then held their nerve for a 4-1 penalty shoot-out victory to reach the last four.
Two goals from West Ham veteran Michail Antonio and one each from Lucas Paqueta and Declan Rice meant that Hugo Cuypers’ opener for away side Gent was irrelevant, with David Moyes side winning 4-1 on the night and 5-2 over two legs.
Gaetan Laborde put Nice 1-0 up and level in the tie after nine minutes against Basel, but Jean-Kevin Augustin levelled late, and Kasim Nuhu scored an extra-time winner.
In the next round, Basel will play Fiorentina, with Alkmaar to face West Ham.