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Leading 1-0 from the first leg, United took control as Marouane Fellaini headed home Marcus Rashford"s cross.
But Celta, needing two goals, levelled on the night through Facundo Roncaglia to set up a tense final few minutes.
United"s Eric Bailly and Roncaglia were sent off after a mass brawl, and the hosts hung on to reach the final.
Indeed they could only celebrate after Celta striker John Guidetti scuffed a golden chance to put the visitors through to their first major European final with the final kick of the game.
The aggregate victory took the Red Devils a step closer to their first Europa League triumph and a return to the Champions League.
They will meet Ajax, who beat Lyon in the other semi-final, in Stockholm on 24 May.
Image caption There were just six seconds left when Guidetti missed the chance to put Celta through
Functional not flashy - but in the final
United had never lost a two-legged European tie after winning the first leg away from home, and knew they would reach the final by keeping a clean sheet at an expectant Old Trafford.
The hosts looked nervy as Celta made an attacking start, before Fellaini settled the tension by converting his side"s first effort on target.
The Belgium midfielder sneaked in at the far post to powerfully head in Rashford"s clipped left-wing cross.
Jose Mourinho"s side created few other chances as they aimed to frustrate the visitors with an organised and largely defensive approach.
Although Henrikh Mkhitaryan, Rashford and Fellaini all drew saves from Celta keeper Sergio Alvarez after the break, it was a functional - not flashy - performance.
Image caption How many Man Utd Euro finals can you remember?
Celta, who had lost their previous five matches and are 12th in La Liga, had more possession than the home side, and forced United keeper Sergio Romero into instinctive saves at the start of each half.
They then ignited the tie with Roncaglia"s glancing header following a corner.
The away goal, which left them needing one more to eliminate United, increased the tension inside Old Trafford, leading to a scuffle involving almost all 22 players on the field.
Bailly was sent off for a swipe at former Manchester City striker Guidetti, with Roncaglia dismissed for retaliating.
With six minutes of added time, there was still opportunity for Celta to create one final chance - but Guidetti blew it.
While United"s players greeted the final whistle with a mixture of relief and jubilation, the visitors slumped to the turf with Guidetti in tears.
"You cannot win a semi-final easily, or be calm. We are in the final, that"s what counts," said United midfielder Ander Herrera.
Image caption United defender Bailly (second left) will miss the final after his red card
Europa League or bust for Mourinho
Winning the Europa League has become Mourinho"s priority as he looks to secure a return to the Champions League in his debut season at Old Trafford.
And the campaign will be deemed a disappointing one if his team lose to Ajax, who edged out Lyon 5-4 on aggregate.
Expectations were high that Mourinho"s arrival would bring an end to the malaise that has surrounded the Red Devils since Sir Alex Ferguson"s retirement.
United have finished seventh, fourth and fifth in the three Premier League seasons since the Scot"s departure - and Mourinho arrived with loftier ambitions than simply scraping into the top four.
Ten months later, the Portuguese has been forced to drastically reassess his ambitions.
He appears to have conceded defeat in the top-four race, with his team sixth, four points adrift of Manchester City going into their final three matches.
With the return leg against Celta at the forefront of his mind, Mourinho made eight changes for Sunday"s 2-0 defeat at Arsenal - a clear admission he felt winning the Europa League was more attainable than breaking into the top four.
However, his decision will only be justified if they are successful in beating Ajax and claiming their sixth European trophy.
Image caption Who are Ajax"s young academy stars making history?
The missing piece in the jigsaw
The Red Devils have won 65 trophies in their illustrious 115-year history, a figure including 20 English league titles and three European Cup/Champions League victories.
However, there is one piece of silverware missing from the Old Trafford trophy room.
In recent seasons winning the Europa League may have been considered an unwanted honour. Now it is considered a must-win as the club looks to complete a clean sweep of the major competitions at home and abroad.
Image caption The front cover of Thursday"s matchday programme creatively illustrated how United hope to complete a clean sweep
Former England winger Chris Waddle on nfnagradi.net Radio 5 live:
"They are not the most technical Man Utd team I"ve ever seen. Mourinho is very good at setting a team up but it is not entertaining.
"United have always been entertainers and even if they win the Europa League, people will say: "Yes they won it, but it wasn"t entertaining."
"The goal for Celta came from United"s lack of concentration and it became a little bit "panic stations" for the rest of the match.
"Ajax are a young team - they could easily freeze in that final."
Manchester United manager Jose Mourinho:
"We were the best team in the first leg but we never kill, we never score goals related to the chances we have. It was an open game at home, all the pressure on our side.
"They were completely free of responsibility and gave us a very hard match. We suffered until the end and it was open until the last second. The boys gave everything they had. I"m really pleased for them.
"After 14 matches, we are in the final. If we win the Europa League, I am more than happy. It would be amazing."
Man of the match - Marouane Fellani (Manchester United)
Image caption Fellaini has regularly been criticised by Red Devils fans since arriving from Everton, but the Belgium midfielder was one of United"s better players against Celta. As well as scoring what proved to be the decisive goal, he impressed in a more advanced role
Before they can fully focus on the Europa League final, United have three more Premier League matches to think about.
They go to second-placed Tottenham on Sunday (16:30 BST), before games against Southampton and Crystal Palace.