Goals from Henrikh Mkhitaryan, Zlatan Ibrahimovic and Juan Mata brought an end to a run of three consecutive league draws for Jose Mourinho's side, who are now two points behind fourth-placed Arsenal.
Leicester had lost their last three games in the top flight and only beaten United once in 17 previous meetings, but Claudio Ranieri's side produced the kind of combative display that underpinned their stunning success last term for much of the first half.
That was until United seized control of the match with two goals in the space of two minutes, Mkhitaryan breaking the deadlock with a powerful run and finish before Ibrahimovic became the oldest player to reach 15 goals in a Premier League season at the age of 35 years and 125 days.
Leicester fans had called for Mata to be sent off for a first-half challenge on Jamie Vardy and their frustrations were compounded when the Spain international made it 3-0 after half-time, having combined brilliantly with Mkhitaryan.
The match became a procession for United in the second half as they responded to weekend defeats for Arsenal and Liverpool to give themselves real hope of clawing their way back into the Champions League places.
For Leicester, a defeat in their 2,000th game in England's top flight leaves them firmly in a relegation fight, with the gap to the bottom three still only one point.
United interestingly opted to match Leicester's 4-4-2 from the start, with Marcus Rashford pushed up alongside Ibrahimovic, and both sides cancelled each other out in a scrappy and uneventful opening 20 minutes.
It was Rashford who missed United's first presentable chance, side-footing over the crossbar after a brilliant run from Mkhitaryan and some good interplay with Mata.
United's frustrations were beginning to show against an initially impressive Leicester rearguard, with Mata a little fortunate to escape with only a yellow card for his late lunge on Vardy, although Kasper Schmeichel made an excellent save low to his right from another close-range Rashford strike.
But just as the teams looked set to go into the break level, United struck in ruthless fashion.
Mkhitaryan netted first, prodding the ball past Robert Huth, racing into the penalty area and firing right-footed beyond Schmeichel despite the goalkeeper getting a touch.
Eighty-eight seconds later, Ibrahimovic doubled the lead, steering Antonio Valencia's low cross from the right back across Schmeichel and into the net after finding clear space 12 yards from goal.
88 - There were just 88 seconds between Mkhitaryan's and Ibrahimovic's goals for @ManUtd. Quickfire.— OptaJoe (@OptaJoe) February 5, 2017
Ranieri brought on Andy King and Demarai Gray at the interval in a bid to inject some life into Leicester's fight, but United needed only another four minutes to quell that resistance.
Mata fed Mkhitaryan on the edge of the area and ran through to meet the Armenia captain's sublime reverse pass, before slotting beneath Schmeichel from six yards.
Riyad Mahrez whipped a free-kick narrowly wide but Mata should have made it 4-0 when he contrived to lift the ball into Schmeichel's grateful grasp after springing the rusty Leicester offside trap.
United knocked the ball around with ease in the closing stages, though Mkhitaryan skied a half-chance to make it 4-0, while Leicester now turn their attentions to a potentially crucial week in which they face an FA Cup replay against rivals Derby County and a league showdown with fellow relegation candidates Swansea City.