Barca were frustrated for the first 59 minutes of the contest, but a red card for Villarreal's Dani Raba on the hour changed the course of the match.
Goals from Suarez and Lionel Messi ultimately secured all the points for the league leaders, but second-half substitute Alcacer also played a key role in securing all three points, according to Suarez.
"A red card always hurts a team, and Paco's introduction went very well. He again showed how important he was for the team," Suarez told reporters.
Barcelona will look to make it back-to-back victories in the league when they host Deportivo La Coruna next weekend.