United survived a frantic finale and a late scare to edge through to the Europa League final on a nervous night at Old Trafford.
Marouane Fellaini's 17th minute header opened up a seemingly comfortable two-goal aggregate lead, but Facundo Roncaglia's 85th-minute leveller left Celta needing one goal to progress.
The centre-back was then sent off in a scuffle with opposite number Eric Bailly - who was also dismissed and will now miss the Stockholm showpiece against Ajax.
The real drama was yet to come, however, when Celta almost snatched victory with the final play of the game.
Former City reject John Guidetti miskicked six yards out with the goal gaping after substitute Claudio Beauvue - who was well-placed to shoot himself - had instead opted to pick him out.
Despite the Reds 100% European home record, Celta started strongly as ex Liverpool man Iago Aspas forced Sergio Romero into a good early save.
A brilliant piece of skill from Paul Pogba almost sent Marcus Rashford clear, but United made the breakthrough they needed on 17 minutes.
Having settled the first leg in Spain, Rashford was again involved as his world-class cross to the back post picked out Fellaini to power in, although goalkeeper Alvarez probably should have done better.
United could have doubled the lead soon afterwards as Rashford burst clear but, this time, he was thwarted by a superb lunging challenge.
Daniel Wass went close with a header and Hugo Mallo flashed a shot across goal as Eduardo Berizzo's side raised their tempo and intensity.
Alvarez saved well from Rashford and Fellaini in quick succession, before Guidetti missed the first of many good chances when he headed wastefully wide.
Romero saved well from Aspas and and substitute Jozabed went close from distance as Celta began to gain the upper hand.
United have made a habit of making things hard for themselves in Europe this season and did so again when Roncaglia glanced home his header from a Pablo Hernandez cross.
That left the visitors requiring only one more to go through on the away goals rule, and tempers flared with the goalscorer and Bailly both sent off in a fiery flashpoint.
Exactly what happened was difficult to see in a melee of players, but Antonio Valenca was also fortunate to escape censure after he clashed with Guidetti.
The former City man endured an eventful night and his misfortune was encapsulated deep into added time.
Celta hoiked the ball up to Beauvue, he spurned a glaring opening to shoot and Guidetti followed up with a hopelessly wayward finish.
Overall team performance: 6/10
United Faithful man of the Match: Marouane Fellaini