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United were beaten in a dramatic penalty shootout after the game had finished 1-1, with David de Gea missing the crucial spot-kick after all 21 other players had found the net.
Solskjaer’s men controlled much of the game but struggled to create any clear cut chances against a resolute Villarreal side, Edinson Cavani’s equalising goal coming only after Marcus Rashford’s shot had deflected fortuitously into his path.
“It’s quiet of course [in the dressing room], its disappointing,” Solskjaer told BT Sport. “That’s football for you, sometimes it’s decided on one kick and that’s the difference between win and losing, we’ve got to learn from that one. We’ve got to - not savour this feeling - but taste this feeling and make sure you don’t get it again.
“We didn’t turn up, we didn’t play as well as we know we can do. We started alright, they got the goal with the only shot they had on target, of course it’s disappointing that we conceded on a set piece.
“After we scored we didn’t control the game and dominate as well as we wanted.
“That’s football at times, they made it hard for us, difficult. We obviously the had the majority of the possession but they defended well and we didn’t create enough big chances.”
It has been a campaign of progress for United, who finished second in the Premier League as well as reaching the Europa League final, but they have ultimately ended trophy-less, with their last piece of silverware coming back in 2017.
When asked whether United could count it as a successful season, Solskjaer added simply: “No.”
The Norwegian did not make his first substitution until deep into the first half of extra-time - by which point opposite number Unai Emery had made five as the balance of play shifted towards his side.
Solskjaer also left De Gea in goal for the penalty shootout, despite the Spaniard not having saved a spot-kick in five years, and when asked whether he had any regrets about his decisions, added: “Now’s not the time to point the finger on what I would have done differently but when you come out with no trophy you’ve not done everything right.”