Jose Mourinho's United will be strong favourites and have the advantage of playing the second leg at Old Trafford as they attempt to lift the Europa League, knowing victory in the final would give them a place in next season's Champions League.
United will travel to Spain for the first leg on May 4 before Celta visit Manchester for the return on May 11.
With United currently four points outside the Premier League top four, winning the Europa League for the first time may represent their best chance of qualifying for the Champions League.
Celta sit 10th in La Liga and progressed from the quarter-finals after following up a 3-2 home win against Genk by drawing Thursday's second leg 1-1 in Belgium.
United have never faced Celta in UEFA competition and the Spanish side are through to a major European semi-final for the first time in their history.
As England's final European representatives this season, United were made to sweat by Anderlecht in the quarter-final second leg on Thursday.
Marcus Rashford's extra-time goal sealed a 2-1 win at Old Trafford that sent United through 3-2 on aggregate.
If United make it to the final in Stockholm's Friends Arena on May 24 it will be the club's first appearance in the decisive match of Europe's second-tier tournament.
It would also be United's first European final since they lost the 2011 Champions League decider against Barcelona at Wembley.
Europa League semi-final fixtures after Friday's draw:
Ajax (NED) v Lyon (FRA)
Celta Vigo (ESP) v Manchester United (ENG)
Lyon (FRA) v Ajax (NED)
Manchester United (ENG) v Celta Vigo (ESP)