After 120 minutes, both teams were locked at 1-1. What followed was one of the most dramatic penalty shootouts in the competition's history.
The result means that Manchester United have now gone four years without winning a trophy.
Even contest decided in dramatic circumstances
Gerard Moreno put the Spaniards in front on the half-hour mark with his 30th goal of the season.
Villarreal remained ahead until the interval. However, United hit back in the 55th minute.
Edinson Cavani showed great striker's instinct as he powered in the equaliser from close range.
Unai Emery made five changes in the second half to try and re-energise his side as they looked for the winner.
Solskjaer, in contrast, didn't make a single change, sticking with his starting XI throughout the 90 minutes.
The United boss eventually opted to make three substitutions in extra time. However, the Red Devils were unable to find the winner.
Both sides failed to conjure up with serious goalscoring opportunities, meaning the final went all the way to a penalty shootout.
Moreno - who scored the game's opener - stepped up first and successfully opened the shootout. Juan Mata provided the perfect response to make it 1-1.
Substitute Daniel Raba put Villarreal ahead. However, Alex Telles converted well to make it 2-2.
Paco Alcacer's shot to the left corner was touched by De Gea but the Spaniard was unable to keep the ball out. Bruno Fernandes suffered a similar fate as his goal made it 3-3.
Alberto Moreno calmly put his spot kick away but the Spanish outfit were pegged back by Marcus Rashford.
Daniel Parejo made it five from five for Villarreal as Cavani did the same for the Red Devils.
Moi Gomez converted Villarreal's sixth spot kick but once again, United replied courtesy of Fred to make it 6-6.
Raul Albiol stepped up and coolly converted to make it 7-6. However, Daniel James showed great composure to level things up.
Francis Coquelin produced the best penalty of the shootout with a powerful effort into the top corner. Luke Shaw replied with a far less convincing effort - but one which ended up in the back of the net.
The drama continued as Villarreal's longest serving player, Mario Gaspar, tucked away his shot. Axel Tuanzebe was able to do the same for the Manchester outfit.
Pau Torres continued the dramatic shootout with another goal before Victor Lindelof levelled up the contest.
Goalkeeper Geronimo Rulli smashed home his spot kick. However, counterpart De Gea couldn't do the same as Villarreal clinched the victory.