Celta goalkeeper Sergio Alvarez had denied the visitors with three superb saves 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, live on Sky Sports on Nissan Super Sunday.
