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.
Newcastle joined their Premier League rivals when Papiss Cisse headed the winning goal against big-spending Anzhi Mackhachkala with the last action of the game to take them through 1-0 after a goalless first leg.
Another wealthy Russian side, 2008 UEFA Cup winners Zenit St Petersburg, also went out to Basel 2-1 on aggregate despite a 1-0 win on the night after they had a late penalty saved against the Swiss team, who played the second half with 10 men.
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.
Fenerbahce drew 1-1 with Viktoria Plzen to go through 2-1 on aggregate after a narrow first leg victory in the Czech Republic and former Europan champions Benfica are also in the last eight thanks to a 3-2 win at Bordeaux for a 4-2 aggregate success.
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.
Spain striker Torres led the way for Chelsea in a rousing second half by scoring a sublime goal, his second in 19 games, and then having to deal with a bloody nose before playing on and striking a late penalty against the bar.
The way Torres and his team-mates battled was typical of Chelsea at present said keeper Petr Cech in the wake of their recovery in an FA Cup quarter-final tie at Manchester United.
"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.
However, Adebayor, who spurned a glorious chance at the end of the first half when Spurs trailed 1-0, got a vital away goal when he slid the ball past Samir Handanovic in the 96th minute after Mousa Demb
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