Despite a 1-0 win on the night in Russia, the defending champions have failed to make it past the last 32 of the competition due to a loss in the first leg of their tie.
Atletico continued to dominate for the remainder of the opening half, but they struggled to create many meaningful chances in front of goal.
Raul Garcia had two opportunities before the break, only to be denied by Sergey Ryzhikov with his first before seeing the second fly wide of the post.
The away side applied plenty of late pressure and they were rewarded for their efforts in the 84th minute when Falcao converted Adrian Lopez's cross.
However, the holders were unable to snatch another goal as Rubin held on for the win.