The England international found the back of the net with a free kick in the second half after drawing a foul from Hugo Mallo.
The hosts carved out the first chance of the evening in the 11th minute when poor marking from United allowed Daniel Wass to get his head to Nemanja Radoja's cross at the far post, but the Dane's effort sailed wide.
Jose Mourinho's side weathered the early storm and took control of the contest, creating their first clear-cut chance in the 19th minute when a curling effort from Rashford was kept out at full stretch by Sergio Alvarez.
They next threatened just after the half-hour mark when a measured through-ball from Paul Pogba sent Henrikh Mkhitaryan clean through on goal, but Alvarez was quick off his line to thwart the Armenian at close range.
With six minutes of the first half remaining, Rashford ventured forward again and teed up Jesse Lingard in the penalty area, but the youngster could only direct his strike at Celta's in-form goalkeeper.
Celta looked more dangerous after the break and they almost carved the Red Devils open when Wass was given too much space in the final third, allowing him to pick out Iago Aspas in the box, but the former Liverpool striker's diving header did not connect.
However, United soon forced Celta onto the back foot again and took the lead in the 67th minute after Mallo chopped down Rashford around 20 yards from goal.
The young forward took a single step to the left before side-footing an effort over the wall, beyond Alvarez and into the back of the net to make it 1-0.
Celta's best chance to draw level went begging in the 73rd minute when Jonny squared a pass into the six-yard box where John Guidetti was lurking, but the striker miscued his connection and scuffed the ball wide.
United will take a crucial away goal into the second leg at Old Trafford on May 11 and face either Lyon or Ajax should they make it to the final.