Celta goalkeeper Sergio Alvarez had denied the visitors three times before he was finally beaten by Rashford on 67 minutes.
The away goal puts United in pole position to advance to their first Europa League final, with the second leg at Old Trafford next Thursday.
Goalscorer Rashford limped off late on and substitute Ashley Young was also replaced after appearing to suffer a muscular injury.
Both may now be doubts to face Arsenal at the Emirates next sunday in Super Sunday fixture.
Paul Pogba returned from a hamstring problem as one of six changes from United’s 1-1 draw against Swansea.
The visitors largely dominated the first half but nearly fell behind after 11 minutes when an unmarked Daniel Wass nodded wide from six yards.
Jose Mourinho’s side then took control and could have been three ahead at half-time but for the brilliance of goalkeeper Alvarez.
He denied Rashford, Henrikh Mkhitaryan and Jesse Lingard before the break, setting the hosts up for a second-half push.
Their main man Iago Aspas was tightly marked for much of the tie but freed himself to flick a header wide shortly after the restart.
Celta came close to snatching the lead when Sisto’s deflected strike from distance drew an excellent stop from Sergio Romero.
The United ‘keeper adjusted superbly to tip the forward’s effort over the crossbar and it proved a crucial save as Rashford grabbed the winner eight minutes later.
He curled home his 11th goal of the season from 22 yards and the first of his United career from outside the box.
But Rashford’s night ended prematurely after he seemingly picked up a knee problem, which places his availability for the Arsenal clash into question.
Celta staged a late rally but failed to test Romero again as United sealed their first knock-out game victory over Spanish opposition since 2008, when they beat Barcelona en route to winning the Champions League.
