Malaga had the better of the opening stages and Roque Santa Cruz failed to make the most of a series of chances, but the game turned on a red card for Darder after 26 minutes for a studs-up challenge on Parejo.
Valencia quickly began to dominate and Alcacer flicked home a cross from Jose Luis Gaya at the near post after 30 minutes.
It got worse for the visitors shortly before halftime when Weligton handled inside the area and the referee pointed to the spot. Malaga keeper Carlos Kameni saved Parejo's spot kick but the midfielder knocked in the rebound.
Piatti then fired home 10 minutes into the second half after a link-up between Sofiane Feghouli and Alcacer. Malaga were denied a consolation when keeper Diego Alves kept out Samuel Garcia's close-range effort.
Valencia reinforced their squad ahead of the season with the expected arrival of Singapore billionaire Peter Lim in the coming weeks and are targeting a Champions League place.
Alcacer's goal capped a memorable day after he was named in the Spain squad for the first time by coach Vicente del Bosque.
"It has been special to be called up and I was like a kid celebrating. Then on top of that it has finished with me scoring a goal," Alcacer told reporters at the Mestalla.
"It is important for me to keep my feet on the ground and work hard each day in training. I am only 21 years old and I still have a lot to improve on if I want to do well in the national team.
"Our (Valencia's) objective this season is to be strong at home and we are a team that has players with a lot of quality that have combined well together."
Earlier, a first-half goal from Alvaro Vazquez gave Getafe a 1-0 victory over visiting Almeria.
- Sports & Recreation
- Carlos Kameni