Eric Bailly put United in front with his first goal for the club late in the first half, during which the home side had defended resolutely.
It was not until the final 10 minutes that the visitors gave the scoreline its convincing gloss, with Romelu Lukaku, Paul Pogba and Anthony Martial all scoring in the space of four minutes.
The result follows a 4-0 victory over West Ham on the opening weekend, while for Swansea the creativity of Gylfi Sigurdsson was sorely missed.
Sigurdsson – provider of nine goals and 13 assists in the league last season – left for Everton on Wednesday in a deal worth a reported £45million following drawn-out negotiations between the two clubs.
The Iceland international had sat out Swansea's season-opening goalless draw with Southampton last weekend, in which the Welsh club attempted just four shots on goal, and their attacking threat was limited once more on Saturday.
Jordan Ayew clipped the crossbar early on but that was one of only six attempts by Swansea – one of those on target.
A new 3-5-2 formation suggested Swansea's priority was keeping their visitors at bay but it was the Welsh club who created the first opening of the match – Ayew's attempted chip kissed the crossbar as it floated across the face of goal.
Lukaku then poked wide after United embarked on a rapid counter-attack, before Phil Jones headed against the woodwork from a Juan Mata free-kick when he should have done better.
Kyle Bartley justified his inclusion as the extra man in Swansea's back line when he made a crucial tackle to halt Marcus Rashford, who would have been clean through on goal without the former Arsenal man's intervention.
An ambitious overhead kick from Martin Olsson – struck more in hope than expectation – was comfortably held by David De Gea midway through the half, but a misjudged Alfie Mawson header then created an opening for Rashford.
The young striker – heckled throughout for the dive that won United a penalty in the last meeting between these sides – lifted the ball harmlessly into Fabianski's arms, before Jose Mourinho's men broke the deadlock on the stroke of half-time.
Pogba's powerful header from a Daley Blind corner was touched onto the crossbar by Fabianski, but Bailly was on hand to finish the job after the ball bounced on the goal line.
The second half was played at a rather pedestrian pace for long spells, as Swansea struggled to find a way through.
When half-chances to level did arise, they were not taken – Tammy Abraham's wayward header on the hour was about as close as they came.
Lukaku then opened the floodgates when he found space in the box to direct a considered shot beyond Fabianski in the 80th minute, before Pogba - who survived calls for a second yellow card during the first half - lifted a delicate chip over the Poland international.
Substitute Anthony Martial then finished the job, cutting inside Mawson to slot home United's fourth.
