The reigning champions battled back twice from a goal down to eventually draw 2-2.
League leaders Valencia went into the clash as one of only two teams still boasting a 100% record this season and they looked eager to continue their perfect start as they twice went ahead early on through an Eric Abidal own goal and Pablo Hernandez.
However, Barca hit back each time with goals from Pedro Rodriguez and Cesc Fabregas to earn a share of the spoils in a contest which both sides will feel they could have won, with Barca looking the more likely scorers in the closing stages.
Valencia finished the match with 10 men following the stoppage time dismissal of Jordi Alba but it was too late for Barca to take advantage.
Gaining a point at the Mestalla is no poor return, but the Barca players largely felt it could have been better.
Alves said: "We are going away with a bittersweet taste. We have the feeling that we did a great job, we made a big effort and the result wasn't enough."
Midfielder Fabregas, whose 77th-minute equaliser took his tally to four goals in four league games since his summer switch from Arsenal, added: "We controlled the second half from start to finish and I think we're going away with a bit of a bad taste in the mouth because we deserved (to win) the game.
"The starts of seasons are difficult but I think we made a great effort, especially in the second half. I think that in the end we deserved the win."
Pedro said on his club's website: "I'm very happy with the goal but it's a shame about the result and for how the game has gone.
"In the first half they started better, but in the second we deserved to win. We've created more chances."
However, not all of the Barca camp felt the three points should definitely have been theirs, with coach Pep Guardiola and his captain Carles Puyol both describing the final result as "fair". Guardiola also said he was "very satisfied with the game".