A goal by Radamel Falcao in the 84th minute gave the visitors hope of overturning the two-goal deficit from the first leg at the Vicente Calderon last week.
However, despite playing with a man extra in the final minutes of the game after Rubin Kazan captain Cesar Navas was sent off, they were unable to score again.
As in the first leg, Diego Simeone's side dominated possession and created far more chances than their counterparts, who were content to sit on their advantage and just focus on defending.
Atleti's surprise 2-0 defeat a week ago meant they faced an uphill task in qualifying for the next round of the tournament before the match had even begun.
Simeone's team selection, containing seven non-regular starters, indicated the holders had given up hope of progressing but his players went all out for the win from the off.
Their first chance fell to Falcao on five minutes, but his shot from the edge of the area was tipped over by Rubin goalkeeper Sergey Ryzhikov.
The Colombian later had a goalbound shot blocked by defender Cristian Ansaldi, just one of many chances Atleti created in the opening half to no avail.
The Spaniards maintained their dominance after the break but created fewer chances, while Rubin striker Salomon Rondon was presented with an opportunity to wrap the tie up but his effort was well saved by Sergio Asenjo.
On the whole, Rubin's chances were scarce as Atleti continued searching for their first goal, which they finally got six minutes from time when Adrian Lopez drove into the left hand side of the penalty area unchallenged and his low cross was slid in by Falcao.
The visitors were left with very little time to get that all-important second goal and their task was made harder thanks to Rubin's time wasting efforts, which saw goalkeeper Ryzhikov pick up his second yellow card in as many Europa League games for the same offence.
Atleti were given a slight lift when Rubin's Spanish captain Navas was given a straight red card on 89 minutes for a reckless lunge on Falcao, but the visitors, who played the entire second half at the Calderon with 10 men, saw out the remaining minutes unscathed.
Over the two games Atletico managed 44 shots on goal but only found the net once, while their opponents scored twice with just 10 attempts.
Rubin can now look forward to a last-16 tie against either Levante or Olympiacos.