Lazio and Fenerbahce, who have both been in trouble with UEFA over fan behaviour, were paired in another tie while Tottenham Hotspur, the only one of the quarter-finalists to have won the trophy, face FC Basel with the first leg in London.
Former European champions Benfica were drawn against Newcastle United, the third English Premier League side still in the competition. None are left in the Champions League.
"It's a great draw for us," Newcastle manager Alan Pardew told the club's official website.
"The Estadio da Luz is a great venue. I've been there and it's a fantastic stadium. Importantly, we are at home in the second leg and that will give us an advantage."
Rubin Kazan chief Kamil Iskhakov said Chelsea should expect a warm welcome in Russia.
"I must say I'm very happy with such a draw," Iskhakov was quoted as saying by local media.
"First of all, I'm a good friend with [Chelsea's billionaire owner] Roman Abramovich from our days together when he was a governor of Chukotka and I was the Russian President's envoy for the Far East region," he said.
"So I'm a big fan of Chelsea, after Rubin of course."
Rubin, who played their two previous Europa League home games at Moscow's Luzhniki Stadium because the grass pitch in Kazan was in poor condition, will host the London club at Kazan's 27,000-seat Central Stadium, Iskhakov said.
Fenerbahce, at home in the first leg against Lazio, had to stage their last 16 second leg against Viktoria Plzen on Thursday behind closed doors after their fans set off fireworks during a previous home match against BATE Borisov.
Serie A side Lazio will play the second leg in an empty stadium in Rome as they complete a two-match sanction for racist behaviour Of fans, poor organisation and firework-throwing during the group stage match against Borussia Monchengladbach.
It was Lazio's fourth racism offence this season.
Draw for the Europa League quarter-finals made in Nyon, Switzerland on Friday:
Chelsea v Rubin Kazan
Tottenham Hotspur v Basel
Fenerbahce v Lazio
Benfica v Newcastle United
First legs to be played on April 4, second legs April 11
