The Spanish side were marginally the better side in the first half and had a chance to take control ahead of the second leg when Rubin's Cristian Ansaldi was given a second yellow card in the 53rd minute.
But they threw away the advantage almost immediately, losing Michel to a straight red card just moments later.
The extra space was not enough to bring a goal, leaving everything to play for when hostilities resume in Russia.
Levante were inches away from the dream start, striking the woodwork after less than two minutes' play.
Sergio Ballesteros was the near-scorer, shooting left-footed from 12 yards after Jose Barkero's corner only to see the ball cannon off the crossbar.
Obafemi Martins had the home side's next shot after eight minutes, failing to worry goalkeeper Sergei Ryzhikov from the edge of the area.
Roman Eremenko was wayward with Rubin's first sight of goal and Garcia Ruben tested Ryzhikov moments later.
Both sides exchanged tame attempts - Salomon Rondon, Papa Diop and Gokdeniz Karadeniz all missing - before Rondon wasted a much better chance from close range.
By the time the half-time whistle arrived, bookings for Rubin pair Eremenko and Ansaldi were the most either side had managed to take from the game.
Keylor Navas, in the Levante goal, could consider himself particularly underworked.
Eight minutes, and one Michel attempt, into the second half, Ansaldi picked up a second caution and was dismissed.
Yet if that decision looked likely to change the face of the game, Levante failed to take advantage - instead evening up the numbers inside just two minutes.
Michel was the guilty party, shown a straight red after fouling Karadeniz.
The absence of two players did not help bring a goal, though, both sides again resorting to 20-yard efforts as Barkero and Rondon both skewed over the bar.
As the game entered its final stages, each side went close to finding the elusive winner and it needed impressive saves from both keepers to keep the stalemate.
Levante were the first to make their move, Martins unleashing in the 79th minute from 25 yards after a period of pressure forcing Ryzhikov to make a fine stop in response.
Rondon fashioned Rubin's response, firing right-footed towards goal only to see Navas beat the ball clear when called upon.