Barcelona finished 13 points behind La Liga champions Real Madrid after ending their turbulent season with a whimper on Sunday by losing 2-0 at home to Villarreal.
Atletico Madrid secured third place by winning 2-1 away at Real Sociedad, retaining their one-point lead over Sevilla, who won 1-0 at home to Athletic Bilbao.
Athletic’s defeat, combined with Villarreal’s excellent victory at Camp Nou, means Villarreal beat Athletic into seventh and qualify for the Europa Conference League.
Villarreal’s season was lit up by their sensational run to the semi-finals of the Champions League but Unai Emery’s team have suffered from inconsistency against the lower sides in La Liga and will be demoted to Europe’s third-tier competition next term.