Having led 1-0 from the first leg, Liverpool were pegged back thanks to Tolgay Arslan's second-half strike on Thursday at the Ataturk Stadium.
When the last 32 tie went to penalties, defender Dejan Lovren was the villain as he skied his spot-kick to end Liverpool's European campaign for another season.
"We're disappointed to go out in the manner that we did. We created opportunities in the first half to make it really difficult for them," Rodgers told reporters.
"Over the course of the game, we defended very well and limited them to very few chances but the one mistake we made, we got punished for, where we just got too deep and they scored.
"We created chances in the first half and in the second half we were comfortable but we were missing a lot of our creativity this evening.
"It's no excuse. There were a lot of young players getting great experience with other players who are creative players. We played with two strikers, so it was an attacking line-up.
"And once Besiktas got the goal, we started to have more of the ball again, but we didn't create so much.
"It was always a tight game, but we just failed to finish the opportunities that we had in the first half."
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