Neville was winless in nine La Liga outings going into the game, but two goals in the space of five second-half minutes saw Los Che ease some of the pressure on the Englishman's shoulders.
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That miss looked like being costly when Espanyol took the lead just a minute later as Oscar Duarte stole in in front of Alves to nod home unmarked from a corner.
Having registered his first ever Espanyol goal, Duarte then threatened to make it an unlikely brace just before the hour mark, but Alves was equal to his effort this time before reacting quickly to deny Victor Alvarez on the rebound too.
Neville turned to Alvaro Negredo off the bench to help him salvage something from the game, and it was a change that paid off when the former Manchester City man levelled things up with a deflected finish after Cancelo had burst into the box.
A second goal arrived shortly afterwards as Valencia turned the match on its head, with Sofiane Feghouli making his way down the right before clipping a cross into the middle for Cheryshev to head home.
Negredo nearly made it three with a header as the hosts looked to put the game to bed, but they also needed Alves in form at the other end to keep out Abraham's effort seven minutes from time.
Gerard Moreno kept the Neville nerves jangling with a miss from close range in the final minute, but Valencia held on for a win that lifts them back up to 11th in the table and leaves Espanyol teetering just above the relegation zone and now winless in 10 matches in all competitions.