Ayestaran is the side's third coach this season following Nuno Espirito Santo's departure in December and Gary Neville's sacking last month.
Valencia's difficult campaign has seen them suffer 13 league defeats as they sit 10th in the La Liga standings with four games remaining.
But after sealing an impressive 2-1 against defending champions Barca at Camp Nou, Ayestaran dedicated the win to the Valencia faithful.
"This is the least that we could give back to the supporters," the 53-year-old said. "We have vindicated ourselves as a team.
"We can be very proud and they deserve this. There are always incentives and every game is an opportunity."
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Valencia were condemned to their heaviest defeat of the season against Barca - a 7-0 loss in the semi-finals of the Copa del Rey in early February - but Ayestaran revealed they did not re-watch the game before Sunday's clash.
"We tried to prepare for today's game with the situations we have right now," he said.
"I think the 7-0 changed Valencia. Today Barcelona could have beaten us. And if they did, our analysis would be different."
Barca's loss - their third-straight in La Liga - saw Atletico join them atop the standings with 76 points.
The defending champions need five wins from their last five games to guarantee the title, and Ayestaran believes they can still go on to win it despite their slump.
"Barcelona is capable of winning five, eight or 10 games straight," he said.
"Now they have some anxiety and they must control it."