Rashford opened the scoring after a brilliant ball over the top from Victor Lindelof set him free to calmly dispatch.
Following a fairly dull affair, a late Bruno Fernandes penalty sealed the victory for the visitors, with two crucial away goals to take home to Old Trafford.
Granada, the underdogs, went into their first European quarter final in their history, while United were making their 27th appearance at this stage.
Ole Gunnar Solskjaer sprung a few surprises with his team selection, with David De Gea taking the number one spot after seven games away from the starting 11, while Rashford beat the fitness test to keep his place out on the left.
Grenada’s team sheet offered a few memorable names, with former Tottenham striker Roberto Soldado and ex- Real Madrid defender Jesus Vallejo making the starting squad.
Story of the Match
The game began surprisingly open with both sides finding success down the flanks in the opening exchanges.
However, following a bizarre appearance from a streaker, the game slotted into the expected flow, with United dominating the ball in front of a wall of Grenada shirts.
Despite this, Grenada’s pressing, particularly from the front three, was admirable and cause the United back line a few issues on occasion.
A fairly dull opening 25 minutes were summarised by just United’s second shot of the night as Marcus Rashford looked to curl one from a tough angle which floated a few yards wide of the back post.
However, minutes later, the game sparked into life as Lindelof picked the ball up from a goal kick before picking out a 70-yard pass over the top to the onrushing Rashford who controlled it excellently into his stride. Another touch set him up one-on-one with the goalkeeper before slotting into the bottom corner with brilliant composure to give United a huge away goal.
The goal came as a breath of fresh air to United, who added a sudden zip and energy to their previously limp attacking display.
The half ended with a flurry of half chances for Grenada, most notably to Soldado, who received the ball unmarked at the back post before controlling with his chest and firing tamely wide of the near post.
The first genuine chance of the half came ten minutes in as Harry Maguire broke from the back line, carrying the ball to the edge of the box before unleashing a shot which swerved a few yards wide.
With the game struggling to break out of first gear, Solskjaer made his second change of the evening, with Edinson Cavani replacing the goal scorer Rashford to minimise the risk of further injury to one of his key players.
The change prompted a tweak in intensity from the hosts thanks to a number of poorly placed United passes as the pressure began to mount.
With 15 minutes remaining, Telles whipped in a delightful cross from deep which Cavani slightly misjudged at the back post as a glorious chance for a second goal went begging.
A second goal became almost imperative as the game went on, with a number of yellow cards meaning that Harry Maguire, Scott McTominay and Luke Shaw all suspended for the second leg.
However, it was Granada that look the more probable, albeit at a stretch, as they began to gain territory into the United half.
This allowed United to benefit on the counter as they have done so many time before, as Dan James picked the ball up on the break, driving at the defence before releasing Fernandes on the overlap. Fernandes took one touch before trying to pick out the far corner from close range, only to be met by the strong arm of the onrushing goalkeeper.
With just moments remaining of regulation time remaining, another Telles cross was this time met by Cavani, however, a whistle sounded as he met the ball to award united a penalty as Fernandes was deemed to have been impeded in the box. Fernandes then stepped up to take the penalty without his trademark hop and fired into the bottom left corner with the aid of a big hand from the goalkeeper, who should have done a lot better.
The goal completed a professional, yet uninspiring victory from United, but a crucial one nonetheless.
Perhaps a negative spin on the game came with the news that three of United’s key players will miss the second leg, however, the two-goal cushion will soften the blow as United have one foot in the next round.
Man of the Match – Harry Maguire
United’s captain played with the aggression and intensity that United fans have been screaming for. He comfortably dealt with everything thrown at him both aerially and on the ground, while his distribution and ability to drive forward caused Grenada problems on occasion.