Leicester, the only English team in the last 16, had a 2-0 lead over Rennes from the first leg but the French club's midfielder Benjamin Bourigeaud pulled a goal back early.
Leicester defender Wesley Fofana headed in an equalizer off a corner kick in the second half in his first appearance of the season after seven months out with a broken leg.
Substitute Flavien Tait put Rennes back ahead in the 78th but the French club was denied a crucial third goal when goalkeeper Kasper Schmeichel pulled off a good save to stop Serhou Guirassy's effort with five minutes left.
Vitesse looked set to take Roma to extra time thanks to Maximilian Wittek’s goal in the 62nd, but Tammy Abraham netted his tournament-high seventh goal of the competition to take the Italian club through.
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Feyenoord added a 3-1 home victory over Partizan to win 8-3 on aggregate.
After a thrilling 4-4 home draw with Copenhagen in the first leg, PSV Eindhoven routed the Danish side 4-0 after two goals from Israeli forward Eran Zahavi.
Marseille came from a goal down to beat Basel 2-1 for a 4-2 aggregate score.
Norwegian champion Bodo/Glimt needed extra time to advance after a 2-1 win for AZ Alkmaar left the aggregate score tied at 3-3. Right-back Alfons Sampsted flicked in a low cross in the 105th minute to send the Norwegian side through.
PAOK won 2-1 away at Gent to make it through 3-1 on aggregate.
Slavia Prague was leading 3-1 at Austria’s LASK but was reduced to nine men after receiving two red cards and lost the game 4-3. But having won the first leg 4-1, the Czech team still progressed.
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