Record Europa League winners Sevilla will not contest the final in their own stadium this season after losing 2-0 after extra-time away to West Ham on Thursday in the last 16.
Ukraine's Andriy Yarmolenko scored the emotional winner in the 112th minute after visiting keeper Bono could only parry a shot from Pablo Fornals.
Earlier Tomáš Souček headed the only goal of an end-to-end 90 minutes to cancel Sevilla's 1-0 win from their home leg.
"We dominated," West Ham's Michail Antonio told BT Sport. "We always believed in ourselves. From the first half we were on them creating chances.
"Andriy Yarmolenko is such a lovely guy. I can't imagine what he's going through. He’s got a finishing touch now. Hopefully he can keep doing it."
There was sensational drama for Eintracht Frankfurt, who reached the quarter-final draw on Friday thanks to Martin Hinteregger converting a free-kick against Real Betis in the last minute of extra-time.
Betis themselves had only gone beyond regulation thanks to Borja Iglesias scoring in the 90th minute to level the tie at 2-2 on aggregate but Frankfurt eventually prevailed.
"The last-minute-Eintract was there again today," Frankfurt coach Oliver Glasner told RTL. "It was an unbelievable game. Amazing success."
Bayer Leverkusen failed to break down an organized Atalanta side as they crashed out of the Europa League following a 1-0 home defeat on Thursday.
The Italians will be joined in Friday's quarter-final draw by Barcelona, who came from behind to win 2-1 at Galatasaray and progress by the same score after a goalless home leg.
Defender Marcão headed the opener for Galatasaray but Pedri levelled before half-time with a with a lovely solo goal.
Pierre-Emerick Aubameyang then bagged the winner five minutes after the restart after keeper Iñaki Peña, who is on loan from Barcelona, made saves from Sergio Busquets and Pedri.
"We played a match of huge personality," said Barcelona coach Xavi. "It was a great performance to fight back in a ground like this against a very good team.
"They knew how to defend against us but we were patient. I'm very happy for my players."
Glasgow Rangers also progressed to the last eight despite losing 2-1 to Red Star in Belgrade having won 3-0 in Glasgow while Braga's 1-1 draw at Monaco gave them a 3-1 aggregate triumph.
RB Leipzig were given a bye to the quarters as Russian opponents Spartak Moscow were banned following the invasion of Ukraine.