United, who won 2-1 on aggregate, will face Ajax in Stockholm on May 24 after the Dutch side overcame Olympqiue Lyonnais 5-4 on aggregate after a 3-1 defeat in Lyon.
Leading 1-0 from the first leg of the semi-final in Spain, United thought they had killed off the tie when Marouane Fellaini headed them in front after 17 minutes.
United sat back and allowed Celta to dominate possession but looked comfortable until Facundo Roncaglia headed the visitors level in the 85th minute, meaning another goal for the Spaniards could have put them through on away goals.
Tempers flared shortly afterwards with Roncaglia and United defender Eric Bailly sent off after a melee near the centre circle at Old Trafford.
John Guidetti had a chance to win the tie for Celta with the last kick of the game but failed to control the ball six metres from goal.
United survived the six minutes of stoppage time to squeeze through on aggregate.
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