The Yellow Submarine are now guaranteed at least a place in the Champions League qualifiers for the first time in four years, but could gain direct entry to the group stages should they go on to win the club's first major honour in the Europa League.
First-half goals from Samuel Garcia and Adrian Lopez - who also struck the stoppage time winner in the 1-0 frist leg success over Liverpool - moved Villarreal six points clear of Athletic Bilbao with two games to play.
Athletic leapfrogged Celta Vigo into fifth as they came from behind to beat the Galicians 2-1 earlier on Sunday.
Fabian Orellana opened the scoring for Celta before Aritz Aduriz equalised with his 34th goal of the season from the penalty spot.
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Gustavo Cabral was then sent-off for kicking Aduriz in the head and Athletic made the man advantage count when Raul Garcia struck the winner 18 minutes from time.
At the bottom of the table, Getafe gave their chances of survival a huge boost with a 2-0 win at 10-man Deportivo la Coruna thanks to goals from Pedro Leon and Carlos Vigaray.
Espanyol moved to the brink of mathematically securing their place in the top flight for next season as Felipe Caicedo's winner handed them a 1-0 win over Sevilla to move five points clear of the relegation zone.