Roma were unable to overcome their two-goal deficit from the first leg as a 2-1 victory at home to Lyon on Thursday was not enough to see them through to the Europa League quarter-finals.
They made the worst possible start, falling behind to a Mouctar Diakhaby strike but Kevin Strootman replied immediately.
A Lucas Tousart own goal on the hour set up a thrilling finale but Lyon held on for a 5-4 aggregate win, sending the French club into the last eight.
Besiktas reached the quarter-finals with a 5-2 aggregate victory over Olympiakos.
The Turkish league leaders drew the first leg in Greece 1-1 but eased to a 4-1 victory in Istanbul despite playing 50 minutes with 10 men.
The hosts quickly built on their away goal when Vincent Aboubakar – Besiktas’ scorer in Athens – netted again to put his side ahead in the tie after only 10 minutes.
When Besiktas extended their advantage through Ryan Babel after 22 minutes the tie looked over.
However, Tarik Elyounoussi pulled a goal back and Olympiakos were given further encouragement just before half-time when Aboubakar, with six goals in European competition this season, was sent off.
But the visitors could not capitalise on their numerical advantage and Babel struck again in the 75th minute before Cenk Tosun completed the scoring with six minutes remaining.
Second-half goals from Hugo Mallo and Iago Aspas saw Celta Vigo win 2-0 at Krasnodar to advance to the quarter-finals with a 4-1 aggregate victory.
The Russian side had Charles Kabore sent off late on.
Genk, 5-2 winners away from home in the first leg, completed a 6-3 aggregate victory against Gent.
Timothy Castagne gave Genk a second-leg lead before Louis Verstraete equalised with six minutes left in a 1-1 draw.
The all-German clash between Borussia Monchengladbach and Schalke finished 2-2 on the night at Borussia-Park, with Schalke going through on away goals after a 3-3 aggregate draw.
It was finely poised at 1-1 after the first leg and it looked like the hosts would progress after goals from Andreas Christensen and Mahmoud Dahoud put them firmly in control at half-time.
But second-half goals from Leon Goretzka and Nabil Bentaleb, from the penalty spot, turned the tie on its head.
Manchester United edged through to the quarter-finals with a narrow 1-0 victory over Rostov.
After a 1-1 draw in the first leg in Russia, Juan Mata scored the only goal of the return match with 20 minutes remaining.
Ajax overturned a 2-1 first-leg deficit against FC Copenhagen with goals from Bertrand Traore and Kasper Dolberg giving them a 2-0 win in Amsterdam and a 3-2 aggregate success.
Frank Acheampong’s second-half goal saw Anderlecht defeat Apoel Nicosia 1-0 and 2-0 overall.