Goals from Juan Mata, John Terry and Fernando Torres, who also missed a penalty, put European champions Chelsea through 3-1 on the night and 3-2 on aggregate after visiting Steaua Bucharest's Vlad Chiriches equalised on the stroke of half-time.
Spurs striker Emmanuel Adebayor pounced in extra time to score a vital away goal in their 4-1 second leg defeat at the San Siro as they scraped past three-times European champions Inter after the nail-biting tie ended 4-4 on aggregate.
Rubin Kazan fared better than their compatriots as they saw off Levante 2-0 in extra time to end Spanish interest in this season's competition.
Lazio's Libor Kozak scored a hat-trick in a 3-1 win over VfB Stuttgart for a 5-1 aggregate victory as they kept Italian hopes alive with a place in Friday's quarter-final draw.
But it was Inter's stirring comeback, spoiled by lacklustre Tottenham's vital away goal, and Chelsea's second-half display to see off spirited former European champions Steaua that lit up a fascinating night of Europa League action.
"We are at the stage where every game is a final," Cech told ITV. "When it went to 1-1 [against Steaua] we had to score twice but at the weekend we managed to get [back from] a 2-0 deficit at Old Trafford so we believed we would get the goals.
"Fernando was working hard and he took his chance well. It's a pity the penalty didn't go in for him but we scored the three goals we needed and we're through."
Tottenham were lucky to continue their European quest after they let Inter come back at them following in-form Gareth Bale's masterclass in the first leg when he scored in a 3-0 win.
"We don't do it the easy way... well done lads," Bale, who was suspended for the Italian trip, wrote on Twitter after Spurs secured a last eight berth by the skin of their teeth.
Inter went into the match after being outplayed in last week's first leg in London but strikes from Antonio Cassano, Rodrigo Palacio and a William Gallas own goal brought them level on aggregate against a shell-shocked Tottenham side.
