Cristiano Ronaldo netted on his return from a club suspension, while Diogo Dalot and Marcus Rashford also registered on a comfortable night for the 20-time English champions.
The Red Devils are now guaranteed to finish at least second in Group E, which would see them enter the knockout round playoffs, but they can still secure first spot in the section, and an automatic spot in the round of 16, by beating Real Sociedad in Spain next week.
Erik ten Hag's side lost 1-0 to Real Sociedad in their opening match in the section, so they will need to better that result in order to advance on the head-to-head record, as the La Liga club still have a three-point advantage at this stage courtesy of their 2-0 win over Omonia.
Ronaldo returned to the first XI after being dropped from the squad against Chelsea as punishment for his refusal to enter the field as a late substitute in last week's game with Tottenham Hotspur, while there was a full first-team debut for 18-year-old Alejandro Garnacho.
After a period of domination in terms of possession, Antony had Man United's first serious attempt of the match in the 10th minute, but the Brazilian's effort was saved by Sheriff goalkeeper Maksym Koval.
Casemiro was next to try his luck, with the midfielder meeting a corner from Christian Eriksen, but towering centre-back Stjepan Radeljic made the clearance.
Brilliant work from Garnacho then created a half-chance for Ronaldo, but Radeljic just beat the Portuguese to the ball, before Man United's returning number seven headed over the crossbar shortly after.
Bruno Fernandes was next to come close, with the Portugal international just missing the post with a clever effort after the home side had managed to break on their opponents.
Ronaldo had another opportunity just before the 30-minute mark when Fernandes headed towards the forward inside the box, but Koval managed to make the save.
Eriksen then saw his long-range strike saved by Koval after smart work from Garnacho, who was impressing on his full debut for the 20-time English champions, before Ronaldo had one deflected just wide.
Tyrell Malacia had a low effort saved by Koval in the 37th minute, before Sheriff had a chance of their own, with Rasheed Ibrahim Akanbi just failing to meet a dangerous cross from the left.
Man United made the breakthrough in the 44th minute of the contest, with Dalot heading an Eriksen cross into the back of the net, as Sheriff's resistance was finally broken late in the half.
Harry Maguire and Rashford were introduced for Lisandro Martinez and Antony at the interval, and the pattern of the match remained the same, with the home side dominating the contest.
Ronaldo had a chance to register just before the hour after working his way into the Sheriff box, but the forward fired his ball wide of the post; the Portuguese did have the ball in the back of the net shortly after, only for the effort to be chalked off for offside.
Rashford headed Man United's second of the match in the 65th minute, with the England international nodding a cross from fellow substitute Luke Shaw into the back of the net.
Koval did well to keep out a low Rashford effort in the 77th minute, before Garnacho just missed the post with a strike from the resulting corner.
The Red Devils had their third in the 81st minute, though, with Ronaldo firing home from close range after seeing his initial header kept out by the Sheriff goalkeeper, with the Portugal international capping his return with a goal.
The returning Donny van de Beek came close to adding gloss to the scoreline in the final stages, but his effort on the half-volley was wide of the post.
Next up for Man United is a home game against West Ham United in the Premier League on Sunday, while Sheriff will be bidding to return to winning ways when they host Dacia-Buiucani in the Super Liga on the same afternoon.