Another Russian club, Zenit St Petersburg, knocked out five-times 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.
"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 t
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