The hosts held the advantage coming into this second leg, having won the first leg in Spain 2-0.
Manchester United manager Ole Gunnar Solskjaer made four changes to the side that won the first leg in Andalusia. Fred, Nemanja Matic, Alex Telles and Mason Greenwood were all handed starts.
A bright start to the game saw the Red Devils launch a devastating counter-attack, but Edinson Cavani's attempted pass to Greenwood was underhit.
The Uruguay international made amends a minute later with a precise left-footed finish. A cross into the box was flicked on by Paul Pogba with the back of his head, and Cavani made no mistake with an emphatic finish from six yards.
Granada went close on a number of occasions but failed to truly test David de Gea, with Manchester United going into the break with a one-goal lead.
The second half saw the visitors rise into the ascendancy, and they imposed their authority on Manchester United, who struggled to string any rhythm together.
However, for all their dominance, they did not make it count in the opposition box. Manchester United seemed content to sit back and try to hit Granada on the counter, with their best chances of the second half coming on the break.
Solksjaer made a raft of changes in the final 15 minutes, with Bruno Fernandes, Aaron Wan-Bissaka and Mason Greenwood all handed rest.
They doubled their lead in the final minute of regular time when Jesus Vallejo headed past Rui Silva under pressure from Juan Mata.
The win sees Manchester United book their spot in the last four, where they will square off against AS Roma. Granada will turn their attention to trying to return to Europe in La Liga.
Here is a rundown of how each Manchester United player fared on the night.
Manchester United player ratings against Granada
David de Gea - 7/10
David de Gea was not really tested on the night, with Granada only forcing him into routine saves. He was assured in his handling and distribution, although he did show hesitancy in coming out on one occasion.
Alex Telles - 7.5/10
Luke Shaw's rejuvenation has relegated Alex Telles to a backup option, but the former Porto man gave a good showing against Granada. He provided width upfront, and it was his crosses that led to both Manchester United goals.
Victor Lindelof - 7/10
Victor Lindelof stood up in the absence of Harry Maguire and excelled in his role as the defensive leader. The Sweden international contributed his quota to Manchester United's clean sheet with six clearances and one interception.
Axel Tuanzebe - 7/10
Tuanzebe was assured alongside Lindelof and played his role in keeping Roberto Soldado quiet.
Aaron Wan-Bissaka - 7.5/10
Although Wan-Bissaka did not offer as much as Telles in attack, his defensive contributions on the night cannot be questioned. The 23-year-old made a game-high five tackles and also had three interceptions.
Nemanja Matic - 7/10
Nemanja Matic was assured in his role as the shield for defense. The former Chelsea man made three interceptions and was pristine in his ball distribution.
Paul Pogba - 7.5/10
Paul Pogba put up another immaculate display for Manchester United. The Frenchman controlled proceedings from midfield and showed his ingenuity in assisting Cavani for the opening goal. He was guilty of overplaying at times, but his halftime substitution led to the hosts losing control of the game.
Fred - 7/10
Fred complemented Pogba in midfield effectively, giving the World Cup winner the license to move forward with the ball.
Bruno Fernandes - 6.5/10
It was a relatively quiet day for Bruno Fernandes, judging by his high standards. The Portugal international was largely on the periphery of the game and failed to make the most of the two presentable opportunities that came his way.
Mason Greenwood - 6.5/10
Mason Greenwood was guilty of indecision in the final third on occasions, while his finishing also let him down.
Edinson Cavani - 7.5 /10
The Uruguay international followed up from his goal against Tottenham with another well-taken strike here.
Donny van de Beek - 6.5/10
Donny van de Beek took some time to get going after coming on at halftime, but he slowly eased into the game. He saw a shot on the hour mark sail narrowly wide.
Juan Mata - 7/10
Mata made his first appearance in over six weeks, and it was his misdirected header that deflected off Vallejo to give Manchester United the second goal.
Amad Diallo - 6/10
Diallo came on with eight minutes to go and had a good chance to run at the Granada defense but fluffed his lines.
Daniel James - 6/10
Daniel James came on on the hour mark and had a pass accuracy of 80%.
Brandon Wiliams - N/A
He did not get involved enough to warrant a rating.