Celta got off to a surging start with plenty of work to do to turn the first-leg result around. Iago Aspas was not given space in Spain, but inside the opening five minutes, he had made two threatening runs down the right flank.
It was a nervy start from the home defence, and the former Liverpool striker tested Romero with a well-struck shot from the right. The travelling fans urged their team on and it seemed to work as they claimed two thirds of possession with Mourinho's team unable to string four passes together.
The Red Devils eventually managed to relieve some pressure with a couple of counterattacks that ultimately came to nothing. But these glimpses of hope were the forerunners of what would truly ease the tension for the hosts.
Rashford curled in a sensational cross to the back post where Fellaini was waiting to head inside the near post. Alvarez got to the ball, but could not do enough to keep it out.
Towards the final minutes of the half, Celta fought back to bring Romero into action. Sisto's curling strike was heading for the top corner before the keeper did well to gather. A few minutes later, the winger whipped the ball in from the left but Wass went for a diving close-range header which whistled past the post.
Celta were on top after the break, but they looked to be missing the cutting edge to really cause trouble for Mourinho's team. The away side enjoyed chances, but Sergio Alvarez pulled off some notable saves to keep his team in the tie.
The home side seemed to have the match under control, but a Roncaglia header five minutes from the end put the pressure back on the Red Devils.
A bust-up involving players from both teams and three of the officials saw Bailly and Roncaglia sent off, which prompted the referee to add six minutes of stoppage time at the end.
Herrera had a goal disallowed for a Pogba foul before Guidetti failed to sort his feet out for an open goal in what was a heart-stopping end to the clash.
Old Trafford cheered as they heard the final whistle, and the Celta players went down to their knees. The away support continued to wave their flags and thank their team long after the end of the game.
Manchester United will meet Ajax in the Europa League final, though Mourinho will be without another defender after Bailly's dismissal. Back to a European final.