Perceived as the minnows of the competition by many, the hosts played with the self-belief of a giant and were inches away from taking the lead when the impressive Yangel Herrera struck David De Gea's left post.
But on a night full of screams from those on the pitch and on the touchline, it was a Marcus Rashford screamer that broke the deadlock at the Nuevo Estadio de Los Carmenes, as the winger raced onto Victor Lindelof's punt over the top and spectacularly plucked the ball out of the air before stabbing beyond Rui Silva.
And just when it looked like the tie was hanging in the balance heading into stoppage time, Bruno Fernandes was deemed to have been fouled in the penalty area. The Portugal international picked himself up to squeeze his spot-kick past his compatriot to give United breathing room heading back to Old Trafford.
That's 20 goals for the season now for Rashford, and if the run was good, the touch was out of this world. United were far from at their best, but they got the job done and the tie is now theirs to lose having banked two valuable away goals.
Ole Gunnar Solskjaer's men were strong favourites going into the tie, but the visitors made a nervy start to proceedings and were guilty of showing a lack of discipline. Luke Shaw, Scott McTominay and Harry Maguire are all out of the return leg at Old Trafford after picking up yellow cards.
Manager rating out of 10
7 -- The Norwegian showed faith in the players who got the job done at the San Siro against AC Milan with an unchanged back four and the inclusion of Rashford. Between them they repaid him with a clean sheet, a remarkable assist and a wonder goal.
Player ratings (1-10; 10 = best, players introduced after 70 minutes get no rating)
GK David De Gea, 7 -- A first start in seven for the Spaniard, who was well-beaten by on-loan Manchester City man Herrera's shinned effort which cannoned back off the post. To his credit, the United goalkeeper showed good hands to hold onto a series of clean Kenedy strikes.
DF Aaron Wan-Bissaka, 6 -- A tough start for the visiting right-back against the unrelenting Kenedy, who pressed hard. However, after taking the time to size up his man he was much more comfortable.
DF Victor Lindelof, 8 -- Did just about enough to close down Roberto Soldado who was given only the side netting to aim at. Then capped a pretty good 30 seconds with a stunning 70-yard pass to pick out the penetrating run of Rashford.
DF Harry Maguire, 7 -- The skipper did well to tidy up a dangerous low ball towards Kenedy inside the first minute, and he looked in the mood at the other end too, turning Carlos Neva inside out on the right flank before lashing a thunderous effort just wide minutes later.
DF Luke Shaw, 6 -- Picked out Fernandes with an excellent ball just before the break only for the move to filter out. His yellow card for a tug on Antonio Puertas means he'll miss the second leg. Made way for Alex Telles at half-time.
MF Scott McTominay, 6 -- Another feisty performance in the middle of the park from the Scot, who did his best to frustrate the spirited hosts and picked up a booking for his troubles.
MF Paul Pogba, 6 -- A forgettable first touch teed up the hosts for a slick counter-attack but his next intervention was much better, charging into the midfield and finding Marcus Rashford with an excellent ball out to the left. Later booked for a clumsy challenge.
MF Daniel James, 6 -- A quiet start from the Welshman who became more of an outlet as the first half rumbled on. His pace was always a threat once United started to move up through the gears, and he played Fernandes through with one neat back-heel, although the flag would have gone up anyway.
MF Bruno Fernandes, 7 -- Nine penalties, nine goals, but few more important than this one as Fernandes' 89th-minute penalty trickled over the line to double his side's advantage. Earlier, he'd made an equally important contribution at the other end, prodding the ball behind with Soldado lurking for a tap-in.
MF Marcus Rashford, 8 -- There were questions over the England man's fitness levels prior to kick-off but he repaid his manager's faith with a superb goal. The perfectly-timed run. The sublime touch. The cheeky poke beyond Rui Silva. It was quite simply gorgeous stuff all-around.
FW Mason Greenwood, 5 -- The teenager did well to quickly move the ball onto Fernandes early on only for Pogba to see his effort blocked, and beyond that he was quiet. He gifted Granada their best chance of the second half with a sloppy pass that invited a counter.
DF Alex Telles, 6 -- Gave the ball away with his first touch before conceding a corner with his second. Beyond that he wasn't really tested as United started to take control once more. Nearly picked out Edinson Cavani with a lovely whipped cross.
MF Nemanja Matic, N/R -- Replaced Pogba for the final 15 minutes or so -- perhaps unsurprising considering the France star was on a booking -- and he put in a decent shift outside of picking up another yellow.
MF Donny van de Beek, N/R -- Came on for the under-par Mason Greenwood for the closing stages as United looked to preserve their advantage.
FW Edinson Cavani, 6 -- Replaced Rashford in the 66th minute. I'm still scratching my head at how Telles' cross somehow evaded him, but he linked up brilliantly with the same man as some excellent build-up play ended with Fernandes' late penalty.