José Mourinho's men didn't have it all their own way, with the home side frustrating them for large periods. But the 20-time champions turned on the style late in the game to cement their status at title challengers.
It's a disappointing blow for the Swans who are now left with just one point from six, but there are positives for Paul Celement to take away and lessoned to be learned.
Here's how we rated each player's performance.
Łukasz Fabiański 6
The Polish stopper was tested as frequently as he might have expected. Fluffed a cross in the first half which could've hurt his side but was unlucky with the opening goal after initially making a superb save onto the crossbar.
Alfie Mawson 6
A fine performance at left centre-back from Mawson constantly frustrated the United attack, staying tight to Juan Mata and denying the Spaniard time and space on the ball, but the travellers' late salvo won't reflect well on the Swansea defence.
Federico Fernández 6
The experienced Argentinian defender was the commanding presence at the heart of the Swans' backline which frustrated the visitors of large periods of the game. Ultimately, he struggled with the movement of United in the closing stages.
Kyle Bartley 6
The powerful centre-back was a dominant presence aerially and stuck to his tasks well; coped United's movement but occasionally troubled by Marcus Rashford when the young England forward look to run in behind. Substituted after 67 minutes.
Martin Olsson 5
The former Norwich City man, playing at left wing-back, was studious and committed in his defensive duties but rarely got forward enough to support this home side's attack.
Tom Carroll 6
Composed and calm in midfield despite being up against the much more physical united duo of Paul Pogba and Nemanja Matić. The former Tottenham Hotspur prospect was not overawed by his opponents for the most part. A late switch to 4-4-2 left him a overwhelmed, though.
Roque Mesa 6
Endured a difficult opening ten minutes or so in his Premier League debut but settled into the game and was tidy on the ball thereafter. More to come form the Spaniard in the future. Taken off after 67 minutes.
Leroy Fer 6
The powerful Dutch box-to-box midfielder was typically energetic and produced a disciplined display on the right of Swansea's midfield trio, even if it meant he was not able to get forward as much as he might have liked.
Kyle Naughton 6
Like Olsson, Naughton was diligent in his defensive work, tucking back alongside the back three, but offered little going forward.
Tammy Abraham 7
The towering on-loan striker impressed with his mature link play and ability to hold off his markers when utilised as the target an in the Swans' attack. Shooting opportunities were few and far between for the youngster but he can nevertheless be happy with his contribution.
Jordan Ayew 6
Ayew offered an out-ball with his pace in the channels and came close to breaking the deadlock in the first half when his swerving cross hit the bar, but was not as precise in his hold-up play as his strike partner.
Luciano Nursing and Wayne Routledge came on after 67 minutes as Swansea switched to 4-4-2, but this ended up contributing to their downfall as United took advantage of the numerical advantage in midfield. Oliver McBurnie was introduced in the closing stages.
David de Gea 6
The Spanish international was seldom tested by the Swans, with Ayew's mishit cross in the first half, with the swerving effort bouncing off the crossbar, the only time he looked in any real trouble.
Antonio Valencia 6
Wearing the captain's armband, Valencia enjoyed a relatively comfortable afternoon at the Liberty Stadium. Not as much of a factor in attack as usual but solid defensively.
Eric Bailly 6
Netted his first goal for the club, knocking the ball over the line after a Paul Pogba header from a corner had been expertly tipped onto the bar by Fabianski. Bailly was typically combative and clean in his defensive duties, although he didn't always get the better of the physical battle with Abraham.
Phil Jones 6
Another solid display. Jones appears to have leapfrogged Chris Smalling and new signing Victor Lindelöf in the central defensive pecking order for the Red Devils. Hit the bar in the first half when he should have scored.
Daley Blind 5
Despite United enjoy the lion's share of possession for large potions of the game, Blind failed to contribute significantly to the Red Devils' attacking play, with his lack of pace not conducive to bursting forward from full-back; seemed to push on more once Anthony Martial was introduced.
Paul Pogba 7
Pogba struggled to influence the game in his usual manner and his frustration boiled over at times. He was decisive ultimately with he powerful header from a corner eventually leading to the deadlock being broken before half-time and sealing all three points late on with a wonderful, chipped finish.
Nemanja Matić 6
Controlled in central midfield, although not nearly as dynamic as in his debut against West Ham last week. Another solid display all the same, though.
Juan Mata 5
The Spanish creator had little influence on the game. Stationed on the right flank, the left footer looked to come inside at every opportunity but served only to clog the central zone when United needed width. Substituted after 74 minutes.
Henrikh Mkhitaryan 6
Starting in his favoured No.10 role, the Armenian playmaker did little to impress, struggling to find space in which to invent and unable to link effectively with his fellow attackers. He grew into the game late on, however, registering his third and fourth assists in two games. Replaced by Ander Herrera with five minutes to play.
Marcus Rashford 7
Greeted by a chorus of boos with every touch, the 19-year-old looked the most likely to make something happen for United throughout. His pace, directness and dribbling skill was a nuisance for the Swansea backline but his finishing let him down. Replaced by Martial after 74 minutes.
Romelu Lukaku 7
Starved of service for much of the game, Lukaku stuck to his tasks in closing down the opposition defenders and exhibited strong link-up play. His effort was rewarded last on with his fourth goal in a United shirt in all competitions.
For the second week running, Martial came on to produce the goods for United, scoring late on and generally wreaking havoc with his pace and dribbling; it's no coincidence that the away side bagged three goals after his introduction. Marouane Fellaini shored up midfield while Herrera had little time to make an impact.