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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.