Marcus Rashford and Bruno Fernandes grabbed the goals as United took a huge step towards the last four.
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Manchester United are in a commanding position in this quarter-final tie after an efficient 2-0 win at Granada in the first leg.
Match in brief
United dominated the early stages but neither side produced any moments of quality before Marcus Rashford broke the deadlock, latching on to Victor Lindelöf's raking pass before prodding his finish past Rui Silva.
That jolted Granada into action with Yangel Herrera hammering an effort against the post and Roberto Soldado dragging a chance wide.
The home side were more adventurous after the interval but United held firm and delivered a telling blow in the dying seconds when Bruno Fernandes was fouled by substitute Yan Brice, the Portuguese dusting himself down to squeeze his penalty past Rui Silva.
Simon Hart, Granada reporter
"Eterna lucha" – eternal fight – is Granada's motto and their characteristic resilience and commitment means they're not a team to throw the towel in, but this defeat leaves them needing a miracle at Old Trafford. They persevered tonight without finding the moment of inspiration needed to break down United's defence, but that late penalty feels like a real sucker punch.
Matthew Howarth, Man. United reporter
Rashford vindicated Ole Gunnar Solskjær's decision to start the forward with a superb touch and finish for United's opener, while Bruno Fernandes's late spot kick surely gives the Red Devils an insurmountable lead going into next week's return leg. United rarely looked like running riot in a game that lacked rhythm and urgency, but it is hard to see a way back for Granada at Old Trafford.
Dan James, Manchester United midfielder: "It's a massive win. We knew it was going to be hard coming here. We had to play our football, which is where the first goal came from. Obviously the second goal gives us a good platform. We need to go into next week with the same attitude as today."
Ole Gunnar Solskjær, Manchester United manager: "It's not a perfect night. We've got three suspensions for the next game but 2-0 is a very good result. We know it's very difficult to get a result like this in Spain. I'm very happy with the result."
Jesús VaIlejo, Granada defender: "We deserved more than this. We prepared for another result but we have to be happy with our performance. If we play like this in Manchester we can win, though I don't know if it will be enough."
- Manchester United have lost only two of their last 28 UEFA Europa League matches.
- The Red Devils have kept four clean sheets in their five UEFA Europa League matches this season.
- Bruno Fernandes has scored 19 of his 20 penalties for Manchester United in all competitions.
- Granada have lost four matches in a row in all competitions for the first time since May 2017.
Granada: Rui Silva; Víctor Díaz, Vallejo, Domingos Duarte (Germán 54), Neva (Foulquier 74); Herrera, Gonalons (Brice 86); Kenedy (Machis 74), Montoro, Antonio Puertas; Soldado (Suárez 86)
Man. United: De Gea; Wan-Bissaka, Lindelöf, Maguire, Shaw (Alex Telles 46); Pogba (Matić 73), McTominay; James, Bruno Fernandes, Rashford (Cavani 65); Greenwood (Van de Beek 85)
Both teams are back in action on Sunday evening, when Granada are away to Valladolid and United travel to London to face José Mourinho's Tottenham. The second leg then takes place on Thursday, with Ajax or Roma awaiting the victors in the semi-finals.