United striker Marcus Rashford struck a stunning second half free- kick to secure the win and put United on course for this month's final, probably against Ajax Amsterdam after the Dutch giants beat visiting Olympique Lyonnais 4-1 in their first leg.
Until the goal United's finishing had been poor with Celta keeper Sergio Alvarez thwarting them several times.
However he was powerless to stop Rashford's 67th minute free-kick.
The visitors had the best chances in the early exchanges, though the Galicians made the first clear-cut opening for the unmarked Daniel Wass who headed wide from close range.
Rashford then forced Alvarez into a flying save from distance as Jose Mourinho's side got into their stride.
Alvarez then thwarted Henrikh Mkhitaryan and Jesse Lingard in one-on-one situations in quick succession before the break as united midfielder Paul Pogba's influence began to grow.
Celta started the second period brightly; with ex-Liverpool striker Iago Aspas heading just wide before Pione Sisto forced United 'keeper Sergio Romero into action with a deflected effort.
Rashford then put United ahead from 25 metres with a curling effort after winning the free-kick himself before gingerly leaving the pitch 10 minutes from time clutching his knee.
To add to United's injury woes, substitute Ashley Young was forced off with a hamstring problem in added time.
Photo: Photosport Marcus Rashford