A goal by Radamel Falcao in the 84th minute gave Atletico hope of overturning the two-goal deficit from the first leg at the Vicente Calderon last week, but that was the end of the scoring as the Spaniards crashed out 2-1 on aggregate.
Simeone's side dominated possession and created far more chances than their Russian counterparts, who were content to sit on their advantage and just focus on defending.
And Simeone could not fault the attitude of his players despite the outcome over the two legs.
"It is important to compete," said Simeone. "Once again, the team showed that it is full of men, with great dignity and commitment to the club.
"We had an absolute intention to play 90 minutes. We failed to turn it round but they did all they had to do to try.
"When this happens it serves to further strengthen the team and the group.
"We just missed chances but were superior over the 180 minutes."
Atletico forward Adrian Lopez echoed his coach's view of the performance, saying: "We have come close.
"We have played the game we had to do. We have been out there attacking, creating chances and we did not have good fortune in some moments.
"When you get knocked out you want to go with your head held high.
"We had a great game in a difficult stadium and we deserved more."