Dejan Lovren’s miscued penalty meant the Turkish side progressed to the last 16 with a 5-4 shoot-out win after a 1-0 victory on the night.
Liverpool had held on at 1-0 in the tie until the 72nd minute, and were eventually outmuscled by a tireless Besiktas.
“We are disappointed of course to go out in the manner we did,” Rodgers said. “We created opportunities in the first half to make it difficult for them.
“I am proud of the team and (there was) a fantastic atmosphere we really appreciate.
“But we are disappointed having put so much into the game.”
Rodgers said Liverpool would now focus on the Premier League, where their campaign has gathered momentum after a dismal start, and the FA Cup where Liverpool face Blackburn in the sixth round.
“We can just now concentrate on our league campaign, which is now going better, and we have an excellent opportunity in other competitions,” added Rogers. “I can only give credit to my players.
“Unfortunately for us Istanbul has not been a happy place for us tonight.”
“I think until they scored the goal we were quite comfortable. Out aim was to keep a clean sheet. We did miss some of our creativity but it’s no excuse.”
Rodgers meanwhile had comforting words for Lovren, who put Liverpool’s fifth and final penalty high and wide of the bar.
“We are disappointed for him as he had a good game. But someone had to miss and unfortunately it was him and us.”