Last season's top scorer Robin van Persie scored twice, as did Danny Welbeck to ensure Moyes' reign started with a win.
Van Persie's contribution was sublime, hooking in the opener before smashing home a 20-yard effort 17 minutes from time, before Wilfried Bony netted a consolation.
Wayne Rooney claimed the assist for the Dutchman's second.
Named as a substitute due to his obvious lack of match practice, Rooney was introduced for the final half-hour and given his desire to join Chelsea, it must have been heartening for the striker to receive a warm ovation from the United fans.
It remains to be seen whether it is enough to change Rooney's view, although given United's steadfast 'not for sale' stance, that may not count for much anyway.
He noticeably did not join in the touchline celebrations after Van Persie scored.
However, he must be aware that even without any significant new arrivals so far, the Red Devils are still destined to be championship contenders this term, and will welcome Jose Mourinho's Chelsea to Old Trafford in nine days' time with many of the fears about Moyes' succession already banished.
After expressing his unease about his side's tough start in midweek, Moyes had more reason to complain at the Liberty Stadium.
His beef turned out to be over nothing more than the booking dished out to Michael Carrick for sliding into Wayne Routledge on the sodden surface.
Sir Alex Ferguson's replacement has estimated it will take him up to 18 months to put his own stamp on Old Trafford.
In the meantime, he has a title-winning squad to shield him from the bedding-in problems so many managerial counterparts experience.
The confidence of winning five championships in the past seven seasons was in evidence throughout the first half as United soaked up pressure, then struck with deadly efficiency.
Swansea had already escaped from a couple of defensive lapses, as Ryan Giggs elected to try and find Welbeck when it seemed more obvious to go for goal himself, before Michel Vorm turned over Phil Jones' volley.
Even Van Persie had failed to take a decent opportunity.
But when the Dutchman evaded the attentions of the home defence and was allowed to drift into space behind the centre-halves because Angel Rangel was playing him onside, there was only ever likely to be one outcome.
And Van Persie showed the class that made him a back-to-back Golden Boot winner, hooking a shot into the corner to make him the first United player to score a league goal for someone other than Ferguson since Peter Davenport.
It is a measure of the former Arsenal man's prolific streak that it was his 75th league goal since January 2011, 26 more than any other player, who just happened to be Rooney.
Almost as well documented has been Welbeck's lack of scoring prowess.
The England man found the net just once in domestic combat last term, but within a minute of Van Persie's effort, he had matched that figure.
Again Swansea were architects of their own downfall as Ben Davies got sucked into a central position attempting to clear a Patrice Evra cross.
That allowed Antonio Valencia to return the ball to the edge of the six-yard area, where Welbeck turned it home.
In his technical area, Moyes created both goals with a punch of the air. Behind him, Rooney watched on, presumably wondering what the future holds as the United board remain resolute in not granting him his preferred move to Chelsea.
It was no surprise Swans manager Michael Laudrup sent a call out for £12million record signing Bony at the break though, the striker having missed out on a starting berth due to midweek international duties in the United States.
Rooney was introduced for Giggs shortly afterwards, a move well received by the visiting contingent, who bellowed the striker's name in their usual manner, with no hint of annoyance.
And Rooney claimed an assist for Van Persie's second, finding his strike partner with a short pass.
Williams paid the price for standing off on the edge of his own box and after Van Persie stepped round Chico Flores he applied a brutal finish from 20 yards that flew into the top corner.
Bony netted a consolation for Swansea but the contest itself was already over, and Welbeck eclipsed last season's meagre tally in stoppage time with a wonderful chipped finish.