Celta goalkeeper Sergio Alvarez kept the Spaniards in the tie before halftime with a string of saves to deny Rashford, Henrikh Mkhitaryan and Jesse Lingard.
However, he was wrong-footed by Rashford’s free kick as United took a giant step towards reaching the final in
The two sides meet again at Old Trafford for the second leg on May 11.
Rashford’s 11th goal of the season was just reward for United’s superiority as they close in on not just the prestige of winning the Europa League for the first time, but also the prize of guaranteed Champions League football next season it would bring.
Celta, by contrast, failed to rise to the challenge of their first ever European semi-final, but they did have the first big chance just 10 minutes in.
Nemanja Radoja’s cross picked out the unmarked Daniel Wass, but the Dane miscued his header horribly wide with just Sergio Romero to beat.
Thereafter, it fell on Alvarez to keep Celta in the game before the break.
His best stop came on 21 minutes when Rashford took aim from just outside the box and his shot was curling into the top corner, only for Alvarez’s fingertips to turn the ball behind.
Pione Sisto shot to fame by scoring twice for Midtjylland against United in last season’s Europa League, but couldn’t repeat the trick when his tame effort was easily claimed by Romero.
Paul Pogba was making his return after missing United’s last two outings, and the world’s most expensive player looked a class apart in midfield.
A powerful run from the Frenchman teed up Mkhitaryan for the visitors’ best chance of the first 45 minutes.
However, Alvarez rushed from his line to block from the Armenian.
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And Alvarez was called into action again moments later to parry Lingard’s imprecise shot after he had been played in by Rashford’s through-ball.
Celta’s top scorer Iago Aspas was anonymous before the break, but started the second half brightly and saw a flicked header float just wide.
Sisto was then denied by a fine save from Romero as the Dane’s deflected shot threatened to loop over the Argentine.
Celta’s greater attacking intent was offering United more space on the counter-attack, and Rashford didn’t make the most of one promising opening after skipping past Gustavo Cabral.
However, the 19-year-old showed his class moments later as he won a free kick on the edge of the area, and then picked himself up to curl home a wonderful effort into the far corner.
Aspas had Celta’s best chance to respond, but fired wide when Jonny’s cross was fired into him at an awkward height.
Celta have already overcome losing the first leg 1-0 at home on their run to the last four with a 2-0 win at Shakhtar Donetsk in the last-32.
But the Galicians need lightning to strike twice at Old Trafford as United look set for their first European final since 2011.