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Another Russian club, Zenit St Petersburg, knocked out five-time European champions Liverpool on the away goals rule while little-known Czech side Viktoria Plzen surprisingly won their tie against Napoli who are riding high in Serie A.
Atletico slumped to a shock 2-0 home defeat by Rubin in Madrid last week and were ousted 2-1 on aggregate even though Radamel Falcao's 84th-minute strike earned them a 1-0 win in the return match in Russia.
"If we had scored earlier we would have given ourselves more of a chance but even then we still had opportunities to get a second goal," Falcao told the club website (www.clubatleticomadrid.com).
"We just didn't have the luck," added the Colombia striker after netting his 25th goal of the season for the Spanish team.
Coach Diego Simeone left a host of regular starters back in Spain for the last-32, second leg tie at a near-empty Luzhniki Stadium in freezing Moscow, suggesting Atletico were focusing more on maintaining second spot in La Liga.
The only blot on the night for Rubin was the straight red card given to captain Cesar Navas in the 89th minute of a match watched by a crowd of less than 10,000.
Rubin will meet another Spanish club, Levante, in the last 16, after the Valencia-based side won 1-0 at Olympiakos Piraeus to knock the Greeks out 4-0 on aggregate.
Champions League dropouts Chelsea, 1-0 up from the first leg, went behind in the 17th minute against Sparta Prague.
Stamford Bridge fell almost silent when David Lafata belted the ball into the roof of the net from close range.
Fernando Torres had another less than distinguished game in attack for Chelsea and produced a hat-trick of near-misses in the first half with his right foot, left foot and head.
The misfiring Spaniard then failed to beat the goalkeeper in a one-on-one situation in the second period before substitute Eden Hazard came to the rescue, cutting in from the left to send a searing left-foot drive into the top corner of the net.
"We left it late and sometimes that's the best way," said captain John Terry after the 1-1 draw gave Chelsea a 2-1 victory on aggregate. "It wasn't our best performance but at least we are through to the next round.
"It was a great goal from Eden. It's great to have players like him on the bench who can come on and turn a game."
Chelsea next play Steaua Bucharest who edged past former European champions Ajax Amsterdam 4-2 on penalties after their clash ended 2-2 on aggregate.
Liverpool's 150th appearance in Europe was an unhappy affair as they went out to Zenit on away goals after the tie finished 3-3 on aggregate.
Luis Suarez netted twice from direct free kicks to help Liverpool record a 3-1 second-leg win in front of a typically noisy Anfield crowd but Zenit hung on to set up a meeting with Swiss club Basel who put out Ukrainians Dnipro Dnipropetrovsk.
"It was absolutely breathtaking how everyone came together tonight," said Liverpool manager Brendan Rodgers of the atmosphere generated by the home fans.
"We were very proud of the club and the players tonight. They showed why they're part of the world's best football family."
Napoli, flying high in second place in Serie A, went out in tame fashion in the Czech Republic, losing 2-0 on the night to Viktoria and 5-0 overall.
Viktoria's goals both came in the second half from Jan Kovarik and Stanislav Tecl. They now take on Fenerbahce after the Turks put out BATE Borisov of Belarus 1-0 on aggregate.
Tottenham Hotspur, winners of the competition when it was called the UEFA Cup in 1972 and 1984, looked to be heading out on away goals against Olympique Lyon until Moussa Dembele struck with a venomous left-foot drive in the 90th minute to secure a 1-1 draw on the night and a 3-2 aggregate triumph.
Maxime Gonalons had put Lyon 1-0 up in the second leg with a deft close-range header following a free kick out on the left from Clement Grenier.
Three Spurs fans were hurt in clashes in Lyon on the eve of the game.
The Londoners now play Inter in a mouthwatering rematch of their 2010 Champions League group stage tie.
Fredy Guarin scored twice and Marco Benassi was also on target as Inter, UEFA Cup winners in 1991, 1994 and 1998, won 3-0 at CFR Cluj to breeze past the Romanians 5-0 on aggregate.
Former European champions Benfica beat Bayer Leverkusen 2-1 in Portugal (3-1 on aggregate) and Girondins Bordeaux progressed with a 1-0 win over Dynamo Kiev in France (2-1 on aggregate).
Big-spending Russian side Anzhi Makhachkala were too strong for Germany's Hanover 96, cruising through 4-2 overall after a 1-1 away draw while Lazio beat Borussia Moenchengladbach 2-0 to go through 5-3 on aggregate.