Liverpool's ambitions of reaching their first European final in almost a decade suffered a blow as Villarreal netted an injury-time winner in the first leg.
Liverpool went closest when Roberto Firmino's low shot hit the post.
The Reds looked set for a creditable draw in Spain until they switched off to allow substitute Adrian Lopez to tap in Denis Suarez's low cross.
Despite the dramatic winner, the Europa League semi-final still remains delicately poised before next Thursday's return leg at Anfield.
The winners will face either Spanish holders Sevilla or Ukrainian side Shakhtar Donetsk, who drew 2-2 in their first leg, in the Swiss city of Basel on 18 May.