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Villarreal goalkeeper Geronimo Rulli kept his composure to score in a nerve-shredding, seemingly never-ending shootout then denied David de Gea as his side won their first major trophy by beating Manchester United 11-10 on penalties.
United's defeat means City are 10 points clear with both sides having three games left to play.
United's win put on hold the title celebrations of champions-elect Manchester City.
Fans stormed the stadium and pitch before kick-off in protest against the club's owners - the Glazer family.
Villarreal face Arsenal in the other semi after the Gunners thrashed Slavia Prague 4-0 away to complete a 5-1 aggregate victory.
United edged their heavyweight tie with AC Milan as Paul Pogba came off the bench to score the goal that secured a 1-0 win in the San Siro for a 2-1 aggregate triumph.
United won thanks to an early penalty from Bruno Fernandes and a second-half strike from Luke Shaw with the result narrowing City's lead at the top of the table to 11 points over United with 10 games remaining.
Chelsea's fifth defeat in eight matches dropped them to eighth place, nine points behind the leaders.
The 33-year-old Uruguayan used the word "negrito" in an Instagram post last month after beating Southampton on November 29, before taking it down and apologising.
Juergen Klopp's side had the chance to open up a five-point lead at the top of the table, but a combination of wastefulness and Newcastle's valiant rearguard kept them at bay.
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