United lost 11-10 to Villarreal on penalties in Wednesday's decider in Gdansk after a 1-1 draw across 120 minutes – the Red Devils ending their 2020-21 season without a trophy.
Solskjaer's United have lost six of their last seven penalty shoot-outs in all competitions, only winning against Rochdale in the EFL Cup in September 2019 during that time.
Edinson Cavani – who cancelled out Gerard Moreno's first-half opener in the 55th minute – became just the third player aged 34 or above to score in a major European final for an English club, after Gary McAllister (36) for Liverpool in the UEFA Cup final against Deportivo Alaves in 2000-01 and Didier Drogba (34) for Chelsea in the Champions League final versus Bayern Munich in 2011-12.
"We've done as well as anyone could imagine," Solskjaer said during his post-game news conference. "We need to do better, we need to work better, harder, cleverer.