The Bundesliga outfit fought back in the second half after going two goals down in the opening 45 minutes, but it was all for nothing in the end as Fernando Torres found the net late on to ensure Atletico took home a two-goal lead.
Schmidt feels Leverkusen were partially to blame for their own problems, but was also quick to highlight Atletico's quality.
"It was a very difficult evening for us. It is easy to see how Atletico have reached two finals in three years," Schmidt said at a media conference.
"We did not start particularly well. We knew of their counter-attacking abilities before the game, but we ended up shipping two goals in the first half. I am proud of our response in the second half, though. We even had an opportunity to make it 3-3, but it was not to be in the end.
"We created our own problems by giving the ball away unnecessarily and not winning enough second balls. We made it easier for Atletico to display their qualities.
"We made some attacking substitutions in the second half which increased our threat in attack, but Atletico are an incredibly dangerous team to play against, particularly when they are winning away from home.
"The quality of our opponents, and our response in the second half, should not be underestimated. To fight our way back into the game with such a young team is extraordinary. You could see just how much life and bravery there is in this side.
"Only Real Madrid have managed to get the better of Atletico in the past couple of seasons, so it would be unfair to say we lost to them because of our lack of experience."