Leading 2-1 from the first leg in Spain, Villarreal put the tie to bed with three goals in the first half, and a late comeback from the Czechs was too little, too late.
Cedric Bakambu scored his third of the tie in the fifth minute before two goals in three minutes at the end of the half from Samuel Castillejo and Bruno Soriano.
Bakambu then netted again in the 49th minute but Sparta kept fighting and had the consolation of goals from Borek Dockal and Ladislav Krejci.
Villarreal are through to a European semi-final for the fourth time and will bid to make their first final.
Sparta knew all about the danger of Bakambu after his two goals in the first leg but he had already had a sight of goal before running onto Manu Trigueros’ perfect through ball, finishing coolly from 20 yards.
Elsewhere, captain Darijo Srna toasted breaking Shakhtar Donetsk’s all-time appearance record by blasting the penalty that sent the Ukrainians into the Europa League semi-finals.
Evergreen Croatia full-back Srna racked up a club-record 486th appearance in Lviv last night, converting a first-half spot-kick and laying on two assists in Shakhtar’s 4-0 victory over Braga.
The 33-year-old defender moved 18th on the list of all-time appearances in European club competition, edging ahead of former France midfielder Claude Makelele — and celebrated in style by firing the penalty that helped cement Shakhtar’s 6-1 aggregate victory.
Srna’s cross later forced Braga defender Ricardo Ferreira into the first of two own goals, as Shakhtar reached Europe’s second-tier semi-finals for the first time since lifting the Uefa Cup in 2009.
The former Hajduk Split defender captained Shakhtar to the 2-1 win over Werder Bremen that secured the club’s first European title, and will now be intent on a repeat seven years on.
Srna capped his milestone masterclass with the cutback assist for Viktor Kovalenko to strike Shakhtar’s third, before Taison’s cross forced Ferreira to turn the ball into his own net for the second time.
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