The victory moved Celta eight points clear of the relegation zone with four rounds remaining in La Liga.
The club parted ways with coach Rafa Benitez earlier this season, replacing him with Claudio Giraldez - who may have done enough to keep them in the top flight.
Anastasios Douvikas scored the winner with a header shortly after coming on as a substitute in the 80th minute, to give Celta their second win in three matches.
Goncalo Guedes had equalised in the 65th minute after Iago Aspas and Jorgen Larsen scored first-half goals for Celta.
Albert Moreno had put Villarreal ahead early on, before the visitors had Santiago Comesana sent off in the 17th minute.
Ayoze Perez and Pablo Fornals scored late in the first half as seventh-placed Real Betis won 2-0 at Osasuna.
It was a third win in four matches for Manuel Pellegrini's Betis as they moved two points behind Real Sociedad, who are sixth and hold the final European qualification place.
Eighth-placed Valencia missed the chance to move closer to the European qualification places with a 1-0 home loss against Alaves.
Sevilla beat second-to-last Granada 3-0 while Rayo Vallecano - who have won just twice in 15 games - lost 1-0 at home to Almeria.
9°C