Barca lost on aggregate to Chelsea on a stunning night at the Nou Camp.
Alves said: "I'd like to thank all the Culés for your support and, at the same time, apologize for what happened. We tired but it wasn't meant to be."
Teammate Cesc Fabregas also said: "It's painful to be eliminated this way, seeing that we had a lot of chances to score.
"It's a very hard blow."