Marouane Fellaini's 17th-minute header put United 2-0 up on aggregate, but Celta hit back late on through Facundo Roncaglia, who was then sent off along with United's Eric Bailly following a centre-circle melee.
United will play a youthful Ajax team in the final at Stockholm's Friends Arena on May 24, when they can complete their trophy collection and -- perhaps more importantly -- book a place in next season's Champions League.
But manager Jose Mourinho will have to do without influential centre-back Bailly, whose suspension leaves Chis Smalling, Phil Jones and Daley Blind as United's only available central defenders.
Instead they have a chance to add a second big trophy to the League Cup won in February and allow Mourinho to end a complicated maiden campaign as manager by winning the fourth major European honour of his career.
Appearing in their first European semi-final, Eduardo Berizzo's Celta gave United several uncomfortable moments and there was panic in the hosts' box during the fraught closing stages.
United started shakily, with Blind looking particularly uncertain, and there were early murmurs of concern around Old Trafford when Iago Aspas cut in from the right and obliged Sergio Romero to tip his rasping shot over.
An audacious juggling show from Pogba roused the crowd and United tightened their grip on the tie when Marcus Rashford's sumptuous cross from the left was headed in at the back post by Fellaini.
But Celta served reminders of their threat as the first half drew to a close, Pablo Hernandez, Pione Sisto and John Guidetti working Romero and Daniel Wass heading wide from Sisto's dangerous drilled cross.
Celta continued to probe in the second half, with Guidetti thrashing inches wide from Sisto's lay-off and heading off-target from Mallo's cross.
United also had their moments, Celta goalkeeper Sergio Alvarez saving from Mkhitaryan and Fellaini and producing a smart block to thwart Rashford after the 19-year-old had jinked his way past two defenders.
Ajax looked to have killed their tie against Lyon following their first leg 4-1 win at home last week when Kasper Dolberg gave them a 27th minute lead.
But a quick-fire double right at the end of the first period from Alexandre Lacazette -- the first from the penalty spot -- gave the French side hope.
Substitute Rachid Ghezzal's heavily deflected header nine minutes from time followed by Dutch international Nick Viergever's sending off gave Lyon extra impetus but they couldn't find an all-important fourth to send the tie into extra-time.
Disclaimer: No Business Standard Journalist was involved in creation of this content
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