The Catalan giants sealed a €20 million (£15.8m) deal for the French centre-back on a four-year contract on Tuesday.
Mathieu's move was finalised following weeks of negotiations between the two sides after the player pleaded with Valencia to sanction a move to the Camp Nou.
When asked about the defender's desire to sign for Barcelona, Salvo told the press: "It's normal, it's human and it always happens in football. It's clear that we would have liked to have Mathieu [for longer]."
Though they have lost the French defender, Valencia strengthened their attacking options by signing Benfica striker Rodrigo on a one-year loan deal.
Salvo is pleased with the transfer and hopes that the 23-year-old will play a big role at Mestalla this season.
"It is a great satisfaction to have a forward with his abilities – young and with his future ahead of him," he enthused.