Leading 3-1 from the first leg at Stamford Bridge last week, Chelsea were given a dream start from Fernando Torres. But Rubin Kazan levelled early in the second half, before Victor Moses put the Blues back ahead.
The Russians then stormed back with goals from Gokdeniz Karadeniz and a controversial Bebras Natcho penalty to win the match. But they fell two goals short of pulling off the ultimate comeback required to reach the last-four of the competition.
Chelsea could have been behind in the third minute when Rubin Kazan worked their way into the area down the right and the ball was pulled back to Roman Eremenko, but his goal-bound strike was blocked by the body of John Terry.
The Blues responded almost immediately with Frank Lampard's ball up the field falling nicely for Torres to coolly lob over the Rubin keeper Sergei Ryzhikov who had a rush of blood to his head and needlessly came charging out of his penalty area.
Chelsea sat back and comfortably soaked up Rubin Kazan's frequent forays into their half of the field, although Alan Kasaev's surging run and pull back teed up Eremenko on 34 minutes, but his shot was well held by Petr Cech.
The best chance for Rubin Kazan came right at the end of the half when Gokdeniz was picked out in the area by Natcho but Cech parried his effort across his area before Chelsea cleared the danger.
The game was all square on the night on 51 minutes when Ivan Marcano got up well to head a Pablo Orbaiz corner past Cech.
However Chelsea responded within four minutes to go back in front when Moses played a one-two with Ramires on the edge of the area before firing a right-foot shot past Ryzhikov.
Rubin Kazan came back again to level when Gokdeniz was left unmarked to head in Cristian Ansaldi's cross from the left from close range on 62 minutes.
Marcano the glanced a header just wide of the far post on 65 minutes before Rubin were given the opportunity to take the lead for the first time on the night when Cesar Azpilicueta was very harshly adjudged to have brought down Aleksander Ryzantsev in the area.
Natcho stepped up to send Cech the wrong way from the spot and left the hosts needing two goals to complete the comeback.
However Chelsea held on fairly comfortably and will go into the draw for the semi-finals in Nyon on Friday.