Forward Alcacer, called up by the European champions for the first time this month, struck with a clinical effort in the 73rd minute after goals from Pablo Piatti and Daniel Parejo had put Valencia well on top at the Mestalla.
Sergio Garcia pulled a goal back for Espanyol from the penalty spot in the first minute of added time but the victory gives Valencia a share of second, level with champions Atletico Madrid on seven points from three matches.
Valencia missed out on a place in Europe last season but are hoping for a revival after years of financial woes once a takeover by Singapore billionaire Peter Lim has been completed.
The club signed forward Alvaro Negredo on loan from Manchester City in the close season and fans are eager to see the 29-year-old in action once he completes his recovery from a broken foot.
Barcelona are the only side to have won their opening three games after they dispatched visiting Athletic Bilbao on Saturday thanks to two goals from substitute Neymar, both assisted by Lionel Messi.
Real Madrid's stuttering start continued when they were beaten 2-1 at home by champions Atletico and Carlo Ancelotti's side are already six points adrift of Barca and four behind their crosstown rivals.
(Reporting by Iain Rogers, editing by Mark Meadows)
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