Marcus Rashford's brilliant free-kick gave United the vital away goal and fired the Reds into the semi final driving seat.
The teenager, who settled the quarter final against Anderlecht, broke the deadlock on 67 minutes here with an exquisite 25 yard set piece.
The hosts were indebted to a combination of excellent goalkeeping from Alvarez and wasteful United finishing - where have we heard that before - to restrict the Reds advantage ahead of next week's Old Trafford return.
The tie is far from over but United's away goal and clean sheet leaves us in a very strong position and firm favourites to progress to the Stockholm final.
The hosts started well in the biggest match in their history and went close after eleven minutes.
Nemanja Radoja's pinpoint cross picked out Daniel Wass, unmarked eight yards out, but he could only send his header wide.
After that early let off, United seized control only to once more come up against an opposition keeper in inspired form.
After Alvarez superbly tipped away Rashford's rasper, he saved well from Henrikh Mkhitaryan and Jesse Lingard twice in quick succession before the break.
Lingard then came within inches of turning home Mkhitaryan's ball across goal, but the Reds dominance went unrewarded as we went in at half time level.
Wass glanced a header wide early in the second half for Celta and Sergio Romero - who had been a virtual spectator up until then - was belatedly called into action.
The Argentinian has proved a more than able deputy for David de Gea in Europe this season and was at his best once more here when he brilliantly tipped Pione Sisto's deflected and dipping effort over the bar.
Then came the decisive moment in the 67th minute when United won a free-kick 25 yards out.
In a move straight off the training ground, Daley Blind shaped to shoot, his decoy run left Celta flat footed and Rashford blasted over the wall and in.
Wass hooked an effort wide and Lingard went close with a difficult chance at the other end when both could have perhaps done better.
The Reds had the chances to put the tie completely beyond the reach of the Spanish side but part one of the job was done with United 90 minutes away from Stockholm.
Overall team performance: 7.5/10
United Faithful Man of the Match: Marcus Rashford