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 said.
"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 Olympiacos 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 but Eden Hazard scored with almost the final kick to scrape into the last 16.
Chelsea next play Steaua Bucharest who edged past former European champions Ajax 4-2 on penalties after their clash ended 2-2 on aggregate.
At Anfield, Zenit hung on against Liverpool to set up a meeting with Swiss club Basel who put out Ukrainians Dnipro Dnipropetrovsk.
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, scraped through with a 1-1 draw on the night and a 3-2 aggregate triumph over Lyon.
The Londoners now play Internazionale 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 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 Hannover, 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.