After coming on at the beginning of the second half at La Ceramica, the Spanish striker had to be replaced by Yeremi Pino in the 78th minute.
"After medical tests carried out on Paco Alcácer, it can be confirmed that the striker suffered a hamstring strain in his left leg during yesterday’s match against Elche CF," a statement read.
"Recovery time will depend on how the footballer evolves."
Unai Emery's men play Qarabag at home on Thursday, with the Yellow Submarine having already clinched a berth to the Europa League Round of 32, and return to LaLiga action on Sunday, away to Real Betis.
Alcacer has made 12 appearances in all competitions this season, scoring 9 goals and providing two assists.