On paper, this looked like a tricky group when it was drawn, but United started the campaign off with a 2-1 victory away to Paris Saint-Germain and looked to take charge of the group with victory over the Bundesliga leaders, who had beaten Istanbul Basaksehir 2-0 in their opener.
After a tight opening to the match, United took the lead via Mason Greenwood's first-ever shot in the Champions League on 21 minutes, after a delightful through ball from Paul Pogba and, following a VAR review for offside, the goal was awarded.
Donny van Beek was named among in the starting lineup and produced a solid display. However, the game changed in United's favour on 68 minutes when he was replaced by Bruno Fernandes.
Fernandes played in Rashford -- who had been introduced five minutes earlier -- and the England man, although the linesman had raised his flag, finished off superbly past Leipzig goalkeeper Peter Gulacsi on 76 minutes.
Again United benefited from VAR as a review showed Rashford was in his own half when Fernandes passed it and the goal stood.
Rashford made it 3-0 four minutes later as he rifled another effort past Gulasci.
The 22-year-old had the chance to seal his first United hat trick when they were awarded a penalty three minutes from time, but he allowed Anthony Martial to step up and score his first of the campaign.
Edinson Cavani had a late goal disallowed for offside before Rashford sealed his hat trick in stoppage time after being let in by Martial.
Rashford is the first United player to score a hat trick after coming on a substitute since his manager, Ole Gunnar Solskjaer, hit four against Nottingham Forest in 1999.
And, while Solskjaer was quick to praise his No. 10, he said his teammates had laid the foundations before he came on.
"Marcus Rashford came on and did well but the groundwork was there from the team, that's what you want that from the subs," Solskjaer told BT Sport. "What a shift they put in.
"Leipzig are a high press, the intensity, so we had to dig deep. "We had to rest a few because the season is relentless, so we used the squad. This group is coming together. The more quality and spirit you have the more you get the performance."
Asked about Van de Beek starting, Solskjaer added: "Donny is fantastic boy. You see the work he puts down."