The Red Devils took a one-goal advantage into the return match after winning in Vigo last week, and Jose Mourinho's side ensured that they would make the final with a 1-1 draw in the second leg, which produced two red cards late on.
The 20-time English champions will take on Ajax in the final of the competition in Stockholm on May 24, with the winner gaining an automatic spot in the group stage of the 2017-18 Champions League.
Man United boss Mourinho made eight changes to the team that started Sunday's 2-0 defeat at Arsenal in the Premier League, with Matteo Darmian, Ander Herrera and Henrikh Mkhitaryan the only three to keep their place.
As for Celta, Eduardo Berizzo's side were unchanged from the first leg last week, which meant that former Manchester City striker John Guidetti once again led the line with support from Iago Aspas and Pione Sisto.
A positive start from Celta saw the away side come close to registering in the fourth minute of action, but Sergio Romero was on hand to make a brilliant save from Aspas, who had danced past Darmian and Daley Blind before curling one towards the top corner.
Celta continued to look the team more likely to make the breakthrough inside the opening 10 minutes, with Aspas and Sisto causing all sorts of problems to Man United's back four, while Mkhitaryan and Jesse Lingard were struggling to get into the contest down the other end of the field.
A brilliant piece of skill from Pogba created a half-chance for Rashford in the 11th minute, but Facundo Roncaglia was across to challenge the 19-year-old inside the Celta box as the game continued to produce exciting moments.
Man United made the breakthrough in the 17th minute when Rashford found a yard of space in a forward position before delivering a sensational far-post cross onto the head of Fellaini, who used his neck muscles to fire the ball past Sergio Alvarez.
The Red Devils might have doubled their lead in the period that followed as Mkhitaryan and Lingard both had efforts from dangerous positions, but the former's volley was over the crossbar, before the latter's effort deflected just over Celta's goal.
Pablo Hernandez brought a smart save from Romero in the 27th minute as Celta looked to hit back, before Sisto tested Man United's number two 11 minutes later as the visitors continued to commit players forward.
The away side's best chance to get on the scoresheet in the first period came in the 42nd minute when Sisto danced into a wide position before fizzing a cross onto the head of Daniel Wass, but the midfielder's effort flew wide of the post as Man United took a 2-0 aggregate lead into the half-time interval.
Celta boss Berizzo looked to his bench at the break, introducing Jozabed in place of Wass, but the same Man United XI were sent out as the likes of Martial, Wayne Rooney and Juan Mata stayed on the bench.
The away side came close to hitting back in the 46th minute when Hugo Mallo drove into the Man United box before flashing a low cross towards the far post, but Romero just palmed the ball away to safety.
Mkhitaryan almost broke the net with a powerful strike down the other end moments later, but Alvarez was on hand to make a super save during an end-to-end period of the match.
Guidetti had a glorious opportunity to register for Celta just before the hour after being found by Sisto inside the Man United box, but the Sweden international fired wide of the post as the home side remained 1-0 ahead on the night.
Rashford might have put the tie beyond Celta in the 64th minute after creating a yard of space inside the penalty area before looking to chip Alvarez, but the Spanish goalkeeper was on hand to make a smart save.
Alvarez made another super stop to prevent Fellaini from finding the bottom corner with a powerful strike 20 minutes from time, although Celta still needed two goals in order to stand a chance of staying in the competition. They almost had one from a long-range Jozabed strike in the 71st minute, but the attacker's effort just missed the post.
Guidetti was next to come close for Celta in the 75th minute after meeting a brilliant cross from Mallo, but the Sweden international headed wide of the post as Man United survived yet another scare.
The pressure from the Spanish side was relentless, and Celta had one back in the 85th minute when Roncaglia headed a Theo Bongonda cross into the bottom corner, setting up a grandstand finish.
Bailly and Roncaglia were then both sent off in the 88th minute after a clash which involved both sets of players in the middle of the park, with Guidetti going down under contact from Bailly, and Roncaglia being handed a red card for his reaction. The pair had to be separated again leaving the field, but play continued into six additional minutes.
Herrera fired home from close range in the fourth minute of added time as Man United looked to book their spot in the final, but the goal was disallowed for a foul as Celta remained in the argument entering the final moments.
Celta had a wonderful chance to break Man United hearts in the 97th minute when Claudio Beauvue found Guidetti inside the box, but the striker could not convert under pressure as the Red Devils squeezed through to the final following an incredible finish to proceedings at Old Trafford.