Argentine Pellegrino was dismissed on Saturday after the side slumped to a 5-2 home defeat against Real Sociedad which left a team expected to be challenging for the top spots in La Liga in mid-table.
"The boss proved his abilities with the work he did with us on a daily basis, but what we worked on during the week didn't happen on the pitch," Valencia's top scorer Soldado told a news conference.
"He had the character and the capability to manage the team but we didn't rise to the challenge or the levels expected for people wearing this shirt.
"Whoever comes in as a replacement, it is the players who have to get us out of this situation."
Pellegrino had guided Valencia into the last 16 of the Champions League and they could still top Group F ahead of Bayern Munich depending on this week's results, but he had only managed five wins in 14 league matches.
It had always been a risky decision to appoint the untried 41-year-old to replace Unai Emery last May, but it was a popular move among fans.
As a player, Pellegrino had helped Rafael Benitez's Valencia win the La Liga title in 2002 and 2004, as well as the UEFA Cup in 2004, and he had worked as an assistant to Benitez at Liverpool and Inter Milan.
His predecessor, Emery, had never really won over the demanding Valencia supporters despite having led a team struggling financially and forced to sell their best players every year to three consecutive third-placed finishes.
Pellegrino was frustrated he had not been given more time by the club, who he felt had bowed to pressure from fans.
"I think it's unfair. It's been a snap decision in the heat of the moment," he told a news conference on Sunday.
"As a player, I endured barracking from the fans on four or five occasions under Benitez and Hector Cuper and both projects ended up being successful.
"We are only four or five points off our objective in La Liga, we're into the Champions League last 16 and still alive in the King's Cup. But this is football."
Valencia visit Lille in their final Champions League group game on Wednesday.
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