Juan Mata struck twice after Valencia teammate Ruben Baraja had his goal cancelled out by Thierry Henry. Samuel Eto'o netted a late consolation for Barca.
Afterwards, Koeman, whose job hung on last night's result, said: "We are in the final of the Copa, with possibilities of winning something important. The players, mainly, and the fans must enjoy this great night for Valencia. As coach I am very pleased with everything after what we have suffered lately."
Dutchman admitted: "It is always important to have opinions in your favor. Soccer is crazy and everything can change in a day. It is always important to have hope of winning something, and if you lose to think about improving in the following game. Now, being in the final of the Copa, we will have more confidence.
"I have always had the confidence on the part of the ex-president (Juan Soler), I also am happy for him, but this can all change quickly again."