Liverpool centre-half Dejan Lovren missed the away side’s fifth penalty to hand Besiktas victory after the Turkish outfit scored all five of their spot-kicks.
It was a tight game but we just failed to take the opportunities we had
Earlier, Slaven Bilic’s side had levelled up the tie on aggregate after Tolgay Arslan broke the deadlock in the 72nd minute to cancel out Mario Balotelli’s penalty in the first leg.
The visitors were without midfielders Philippe Coutinho, Steven Gerrard and Jordan Henderson, and Rodgers admitted the Reds missed their trio of playmakers after just seven attempts on goal.
“We are disappointed of course to go out in the manner we did,” Rodgers told his post-match media conference.
“I thought we created opportunities in the first half and over the course of the game we defended very well. We just got too deep and they scored.
“I am proud of the players, the young players have had a great experience in a fantastic atmosphere.
“We were missing a lot of our creativity this evening but there is no excuse.
“We played a lot of young players, we played with two strikers, so it was an attacking line-up and then once Besiktas get the goal we didn’t create so much.
“It was a tight game but we just failed to take the opportunities we had.
“When you are away from home the objective is to keep a clean sheet.
“Our objective this season was to win a cup. It is not to be in this competition but we have an excellent opportunity in another competition [the FA Cup].”
Liverpool have little time to recover from their European exit ahead of the visit of defending Premier League champions Manchester City in Sunday’s early kick-off at Anfield.
The Reds are currently in sixth place in the table, two points behind fourth-placed Manchester United.
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