The young striker, who scored United's winner in the quarter-final, was once again the difference for Jose Mourinho's men as his brilliant free-kick in the second half settled a game that the Red Devils dominated for large parts.
The home side will be very disappointed with their performance during the game but they won't lose hope as they will feel with only one goal in the game, they can score at Old Trafford next week to get themselves back in the tie.
Both managers made plenty of changes to their starting lineups
Both managers made changes to their lineups for the game with the home side making nine changes from their weekend game with their top scorer Iago Aspas coming back into the team.
Hosts started strong but tailed off as the half went on
It didn't look that way at the start though as the home side started well as they pressed from the front and they almost took the lead early on.
Nemanja Radoja played a wonderful floated cross into the box to Daniel Wass but the winger put his header well wide of the goal with only Sergio Romero to beat.
That chance seemed to wake the visitors up and they almost took the lead themselves when Rashford hit a brilliant curling effort towards goal but Sergio Álvarez made a brilliant save.
Same old story for United as they miss plenty of chances
From their United took control of the game but once again it was the same old story as they missed some really good chances as they have done all season.
The best of the lot fell the way of Henrikh Mkhitaryan after Pogba played a wonderful ball through to the midfielder but his shot towards goal was brilliantly saved once again by Sergio.
Not long after that chance, good link up play from Jesse Lingard and Rashford played the winger through on goal but he also saw his goalblund shot saved by the keeper.
Therefore despite their dominance after a slow start, United went in at the break knowing that them chances could come back to cost them if they didn't improve their finishing in the second half, while the home side knew that they will would create more chances themselves.
Rashford's brillance in the second half settled the game for Mourinho's men
After the restart, the Sky Blues started strong and tried to take the game to United more and they almost got the opening goal after Pione Sisto saw his shot brilliantly saved by Romero after taking a deflection off Antonio Valencia.
Apart from that chance though the home side really struggled to create many clear cut chances against a very stubborn United defence on the night.
Mourinho's men needed a bit of magic from someone in the second half after not doing much and that is exactly what they got with their first chance when a brilliant free kick from Rashford went into the top corner of the net with Sergio well beaten in the goal.
It should have been two soon after when the ball fell to Lingard in the box but with the goal at his mercy, he blasted the ball wide of the goal when he should have done better.
More injuries for Mourinho to worry about
Before the end of the game Rashford and Ashley Young, who came on as a second half substitute, both left the field injured to leave Mourinho with more selection headaches ahead of the game against Arsenal at the weekend.
Overall though United now will be confident of finishing the tie off in the second leg at Old Trafford next week, while the Spaniards will need to play better to get themselves back into the tie.