The game at Old Trafford finished in a 1-1 draw but United went through after winning the first leg 1-0 and things got even better for them on the night when Marouane Fellaini headed home 17 minutes into the game.
The visitors refused to give up, and were allowed to dominate proceedings by a retreating United side, grabbing themselves an equaliser five minutes from time when Facundo Roncaglia headed home to set up a tense finish to the game.
Despite a golden opportunity for John Guidetti in the final moment, Celta fans were left in tears along with their players as they couldn't find a second goal. Both sides ended the game with 10 men as Eric Bailly and Rongalia saw red in a brawl which summed up a very even game in Manchester.
Both managers made plenty of changes to their team as the visitors started the game strong
As expected before the game both managers made many changes to their teams after resting a lot of players for their respective games at the weekend. José Mourinho fielded the same starting eleven that started in Spain, which is exactly what the visitors did as well.
Going into the game the visitors knew that they needed to play much better than they did last week and it was shown early on as they were pressing the home side high up the field and that almost saw them take the lead.
Iago Aspas went close when he forced Sergio Romero into a great save from outside the box which should have served as a warning to the hosts.
Fellaini headed home for the hosts with their first attack
It did look like it as they took the lead with their first real attack of the game when a brilliant cross from Marcus Rashford found Fellaini unmarked in the box and the midfielder made no mistake by heading home a very important goal for his side.
The goal, though, didn't stop the away side from going for it and they went close on another couple of more occasions before the half-time whistle.
Pedro Hernández forced Romero into another very good save while Daniel Wass, just as he did last week, headed wide from close range after Pione Sisto played a fantastic cross into the box.
However, though, the Red Devils did see the game to the break with the lead but knew that they couldn't switch off at all during the second half to allow the visitors back into the game.
Alvarez was sensational once again for the visitors with vital saves
United started the second half much better than the first and almost got a second goal when Henrikh Mkhitaryan's fierce shot was brilliantly saved by Sergio Alvarez.
Rashford was next to go very close to scoring when he was played in on goal but he saw his chipped effort well-saved once again by the impressive Alvarez.
From there, though, the home side started to sit back rather than taking the game to the visitors and that was almost their downfall when a brilliant cross from Hugo Mallo found John Guidetti unmarked in the box but he put his header wide of the goal when he should have done better.
Roncaglia equalised late on for the visitors as both teams finished with ten men
Mourinho's tactics in the game came even more into question when his team did indeed concede an equaliser five minutes from time. Roncaglia headed home from a cross to set up a very tense finish to the game.
It became even tenser when both teams were reduced to 10 men after a massive brawl in the middle of the pitch. Bailly was sent off for United while Roncaglia saw red for his reaction as well.
Then right at the end of the game with the last kick of the ball and with the goal at his mercy, Guidetti mishit his shot to allow United to advance to the final in a game that they will feel they just did enough to get through.