Diego Martinez's men were full of energy and were aggressive in their approach, committing 14 fouls in 90 minutes to go with 24 tackles, which is over thrice the tally attempted by the Red Devils. Ironically, it was the visiting side that left the stadium with five of their players booked by referee Artur Soares Dias on the night.
It was far from a vintage display from the Red Devils, who had to wait until the final minute of regulation time for their second goal of the night after Rashford's brilliant opener broke the deadlock in the 31st minute. Second-half substitute Yan Brice Eteki's foul handed Bruno Fernandes a spot-kick in the 90th minute, which the Portuguese converted despite his compatriot, Rui Silva, going the right way.
The victory put Solskjaer's men in a very strong position to progress into the semi-final as they took home a two-goal advantage with them to Old Trafford, leaving Granada with a mountain to climb.
Here are some of the major talking points from the game.
#5 Suspensions aplenty for Manchester United
The first half of the game saw a total of three yellow cards to Manchester United, the most they've received in Europe this season within 45 minutes. Paul Pogba was the first to be booked for his reckless tackle inside the first ten minutes.
As Granada ramped up their search for an equaliser after Rashford's opener, a strong spell for the hosts saw Luke Shaw and Scott McTominay pick up yellow cards in quick succession towards the end of the half. Both players are set to sit out the return leg due to suspensions. The Englishman was hooked at half-time, and McTominay — who got away with a rather dangerous challenge inside the box in the second half — was allowed to see out the 90 minutes despite his booking.
The second half saw captain Harry Maguire booked as well despite protests of his foul on Roberto Soldado being his first of the night. Maguire will also miss next week's game, ensuring that Granada have plenty left to play for. Substitute Nemanja Matic was also booked for his first offense since coming onto the pitch, making it a total of five Manchester United players to be booked on the night.
Including two yellows for Yan Brice Eteki and Domingos Duarte, a total of seven yellow cards were dished out on the night.
#4 David de Gea's return to action
One of the biggest talking points for Manchester United ahead of the summer window is the dynamic between their two star goalkeepers. Academy graduate Dean Henderson has capitalised on David de Gea's questionable displays by displacing the Spaniard as Manchester United's number one keeper, much to De Gea's dismay.
On Thursday night, De Gea was called upon to start for the Red Devils by Ole Gunnar Solskjaer, and the 30-year-old helped Manchester United to their third successive away clean sheet in Europe. It was the veteran's first start for the club since their 0-0 draw against Chelsea in late February.
However, De Gea's involvement against Granada raised more questions than it provided answers. With Dean Henderson retaining his spot in the Premier League, is De Gea now established as the Red Devils' cup goalkeeper? Or has Solskjaer opted to trust De Gea's experience in bigger games by starting him in a European quarter-final?
With the summer transfer window just around the corner, rumours have suggested that Manchester United would like to offload their four-time Player of the Year and his £375,000-a-week wages. It remains to be seen if De Gea will regain his place as the Red Devils' number one keeper, play second-fiddle to Dean Henderson, or seek new pastures.