Mason Greenwood scored his first goal in the competition to give United the lead in the first half with a fine finish from a tight angle at Old Trafford on Wednesday.
But it was fellow striker Rashford, a 63rd-minute substitute, who left his mark on the game after twice breaking clear of the Leipzig finish and finishing emphatically.
Anthony Martial was on target from the penalty spot before Rashford put the gloss on the scoreline with another goal late on.
The victory leaves United top of Group H on six points, three clear of Leipzig and Paris Saint-German, who were 2-0 winners over Istanbul Basaksehir.
Leipzig headed into the game top of the Bundesliga after an unbeaten start to the season in all competitions but were comprehensively dismantled by the Red Devils.
Ole Gunnar Solskjaer's side took a first-half lead with a goal which owed much to the dynamic play of Paul Pogba after fellow midfielder Fred won the ball with on the left flank.
The France international left four Leipzig's players in his wake as he burst forward from midfield and slipped a clever pass into Greenwood, who clinically guided an angled finish into the net, the goal awarded after a tight VAR offside check.
It meant Pogba registered an assist in back-to-back Champions League games for the very first time.
Leipzig dominated the remainder of the half, but the game was quickly beyond them when Rashford replaced Greenwood just after the hour.
Rashford netted his first when he latched on to a pass over the top from Bruno Fernandes and broke clear of the Leipzig defence before slotting home – the goal given after a VAR check deemed his run had begun in United's half.
The England striker scored a second with an emphatic low finish after United stole possession high up the field before the home side were awarded a penalty when Martial was tripped.
Rashford declined to take the spot-kick – allowing Martial to register his first goal of the season – but he still had time to complete a treble in the second minute of injury time, drilling home clinically.
What does it mean? United mean business
A surprise away victory over PSG in their opening game signalled United's intent in the competition and this was a real statement.
Leipzig had not been beaten this season, so for United to record their biggest Champions League victory since 2013 could be a significant moment.
With consecutive fixtures against Istanbul Basaksehir to come Solskjaer will have one eye on wrapping up qualification for the knockout stage with two games to spare.
Rashford oozes class on and off the pitch
Rashford became only the second United player to score a hat-trick as a substitute after Solskjaer's treble against Nottingham Forest in February 1999 in the Premier League.
The United striker has grabbed the headlines off the pitch in recent weeks, but his football stock will be just as high after such a clinical exhibition of finishing.
Van de Beek fails to take his chance
With only two starts in the EFL Cup Donny van Beek finally got a start in a competition which really matters, but failed to make a significant impact in an advanced midfield position, creating two chances.
United were far more effective when Fernandes joined the fray and it is hard to see when the signing from Ajax will displace his rival.
Key Opta facts
- Leipzig conceded more goals in this game than in their previous seven matches combined in all competitions this season.
- United have kicked off a Champions League campaign with back-to-back wins for only the second time since Alex Ferguson left the club in 2013, also doing so under Jose Mourinho in 2017-18.
- Despite only making his debut in February, no United player has registered more assists in all competitions than Fernandes (12) under Solskjær, who took charge in December 2018.
- Greenwood has scored six goals in major European competitions as a teenager, the joint-most of any Englishman alongside Rashford.