Celta Vigo boss Eduardo Berizzo said he was proud of his men after what was the Spanish club's first major European semifinal, but admitted it was "a shame" not to reach the final.
Marouane Fellaini's first-half header doubled United's aggregate lead before Facundo Roncaglia set up a nervy final few minutes by nodding home for the Spanish side.
Eric Bailly and the visitors' goalscorer were later dismissed before Celta striker John Guidetti wasted a golden chance in the final moments of the match.
And Berizzo told his postmatch news conference he was pleased the way his team performed on a bittersweet night at Old Trafford.
"It was a great opportunity as it was right at the end but we created half-chances right throughout the game," Berizzo said.
"We played really well and reduced them to one chance, from which they scored. We had a lot of possession, played exactly as we wanted to play. With all the effort we put in, it was a great shame not to reach the final.
"Football's full of drama. You learn from defeats and turn them into victories down the line.
"I couldn't be more proud of my players. They covered every blade of grass and the connection they made with the fans who travelled from Galicia, I'm proud of all that.
"I think, when you analyse it in the cold light of day, it will confirm what I thought about my team all season. We were brave and we fought to the end and I'm proud of them.
"We performed much better here than United did in the first leg. It's a good way to compare the performances. Given that, we perhaps deserved to go through.
"I'm gratified by many things I saw. The fact that we played better here than they did in Balaidos gives us the advantage in overall performance.
"We left everything out there. We couldn't have played any better, I don't think. Given the huge gap on every level between the two clubs, I think with our fight and spirit we managed to bridge that gap.
"Even through the bitterness of defeat, when you settle down then you learn from the bad experience of defeat."