First-leg matches are played on Feb. 14, and return games on Feb. 21.
UEFA also drew the last-16 pairings, putting Inter and Tottenham on course to meet again after thrilling Champions League group-stage matches two years ago. Chelsea and Ajax could meet in March if both win through their last-32 matches.
Zenit, which also started in the Champions League, was content to get an attractive opponent.
"Liverpool is one of the best European clubs. It is one of the most difficult draws but we are happy," Zenit chief executive Maxim Mitrofanov told The Associated Press.
The first leg in Russia will likely be played in around minus-10 Celsius temperatures, though Mitrofanov played down the potential advantage.
"We are also a very technical team so we also need a good pitch," he said.
Atletico seeks a third title in four seasons in the Europa League, which has often struggled in the shadow cast by the Champions League.
"We're happy to be champions," Atletico director Clemente Villaverde said. "I think there are very, very important teams in it. Rubin Kazan is a very difficult team."
UEFA President Michel Platini showed his support by meeting with officials from the 32 clubs, arriving after the Champions League last-16 draw ceremony was completed.
Inter ambassador Luis Figo also defended the competition, which this season offers one-sixth of the prize money paid to clubs in UEFA's signature event.
"Of course it's a prestigious competition for us," Figo told AP. "You just have to look at the teams in the competition and see it's difficult to reach the final."
The Italian club's route to the final in Amsterdam will have to go through England or France.
"Lyon are a club with many trophies and are a top team," Tottenham coach Andre Villas-Boas said. "It's a huge fixture and will fuel excitement."
Tottenham's group-stage rival Lazio, which was paired with Borussia Moenchengladbach, still awaits details of UEFA disciplinary charges after Italian fans directed abuse at the English club's traditional Jewish following inside the Olympic Stadium in Rome.
UEFA general secretary Gianni Infantino warned clubs to curb unsporting and racist behavior: "Don't let it spoil the party," he told officials.
The Zenit vs. Liverpool matches are a likely focus for debate about racism.
Some of the Russian clubs fans this week urged the club not to sign black or gay players, while Liverpool forward Luis Suarez served an eight-match ban last season for directing racial comments at Manchester United defender Patrice Evra.
"Yes, we probably have some fans who have different opinions about racism," Mitrofanov told AP. "But we are here playing football and we are against racism. Playing in this competition will help."
- AP
