The Colombia striker's late strike was not enough for Diego Simeone's side in Moscow and even the late sending off of Cesar Navas failed to spoil the home team's night
The Colombia international slotted home in the final 10 minutes but it was not enough to stop the Spanish side from being beaten 2-1 on aggregate.
Cesar Navas was sent off late on for a tough tackle on Falcao, but Atletico couldn't make their numerical advantage count and must now concentrate on securing qualification for the Champions League in La Liga.
Needing at least two goals to give themselves a chance of progression, the visitors started in positive fashion, and Falcao needed less than three minutes to test Sergey Ryzhikov as his effort from the edge of the area was tipped over.
The Russian outfit were happy to defend in numbers after their victory in the first leg, and despite Cristian Rodriguez testing Ryzhikov from all of 40 yards with a swerving effort, they were comfortable in dealing with the Atletico threat.
The home side did have a chance of their own when Roman Eremenko shot straight at Sergio Asenjo, but Diego Simeone's side remained the more adventurous and Raul Garcia was the next to try his luck from distance, but his effort flew just wide of the post.
With just under 10 minutes until the break, the Liga outfit were almost handed the opening goal as Ryzhikov flapped at a cross, and with the ball bobbling goalwards after bouncing off Sergei Kislyak, Oleg Kuzmin hacked it away off the line.
Garcia headed straight at the goalkeeper from the resulting corner as the pressure increased on the home side's goal, but they could not break the deadlock before half-time.
Former Spain Under-21 international Garcia was looking the most likely for Atletico, and he again managed to fire off a shot from distance early in the second period, though it never looked like troubling Ryzhikov.
Rubin were very happy to sit on their lead, though they nearly put the tie beyond doubt as an error from Cata Diaz allowed striker Rondon a clear run on goal, but Asenjo saved well with his feet.
With the match entering its final 30 minutes, teenager Saul Niguez burst through the hosts' defence, but just as he shaped to shoot, Navas slid in and pulled off a superb block.
The visitors did eventually get their goal with seven minutes left on the clock, as the much-coveted Falcao squeezed Adrian's cross from the left just under the body of Ryzhikov's from close-range.
Navas was given his marching orders in the final minute of normal time after a tough challenge on Falcao, but the Russian outfit held out to reach the last 16 of the competition.