Gomis opened the scoring in the ninth minute with a composed finish before Ayew, who was also on target in the opening-day 2-2 draw at Chelsea, scored with a powerful second-half header after Newcastle full-back Daryl Janmaat had been sent off for a second bookable offence by referee Mike Jones.
Janmaat was booked after clattering into Jefferson Montero twice and the Dutchman's luck ran out in the 41st-minute when he pulled the Ecuadorian back to stop him launching a Swansea attack from deep.
The defender's departure left Newcastle with a mountain to climb and Steve McClaren's decision to withdraw Moussa Sissoko - the Magpies' best player in the first half - and Papiss Cisse revealed a preference for damage limitation rather than offering any real hope of getting back into this Barclays Premier League contest.
Swansea began with a swagger befitting a side who had held the champions on the opening day of the season, with Jonjo Shelvey and Gylfi Sigurdsson both testing Newcastle goalkeeper Tim Krul from distance.
The breakthrough soon came as Shelvey threaded a delicious ball through for Gomis to race onto.
Newcastle waited in vain for an offside flag as Gomis showed excellent composure to round Krul and score
Swansea were playing some fluid football but the goal failed to knock Newcastle out of their counter-attacking stride and Lukasz Fabianski had to be alert at his near post to turn Gabriel Obertan's low shot aside.
Sissoko also found Obertan in a threatening position and, although the Frenchman's first touch was heavy, the ball was cut back with only intelligent defending from Federico Fernandez preventing Cisse from equalising.
Newcastle had edged their way back into the contest but things got a whole lot tougher for them when Janmaat saw red for a dozy piece of defending as he pulled back Montero deep inside the Swansea half.
Montero almost immediately capitalised on his extra freedom when he cut inside stand-in right-back Vurnon Anita and Sissoko to thud the Newcastle post from 20 yards.
Swansea struck the woodwork again within three minutes of the re-start when Sigurdsson hit the underside of the bar from 20 yards but the second goal soon came.
This time Montero raced beyond Chancel Mbema and his cross was met by the powerful head of Ayew, who gave Krul no chance to register his second goal in English football after his debut strike at Chelsea.
Ayew almost added another but Swansea were in cruise control as they made it six wins out of nine against Newcastle in the Premier League.
TWEET OF THE MATCH
"Steve McClaren's come down to the touchline
No need for a brolly though, lovely day at the Liberty." - PA's Phil Blanche on the former England boss.
PLAYER RATINGS
Swansea
Lukasz Fabianski: 7 (out of 10)
Kyle Naughton: 7
Federico Fernandez: 7
Ashley Williams: 7
Neil Taylor: 7
Jack Cork: 7
Jonjo Shelvey: 8
Gylfi Sigurdsson: 7
Jefferson Montero: 9
Andre Ayew: 7
Bafetimbi Gomis: 8
Subs
Nathan Dyer: 6
Wayne Routledge: 6
Leon Britton: 6
Newcastle
Tim Krul: 6
Daryl Janmaat: 3
Fabricio Coloccini: 6
Chancel Mbema: 7
Massaido Haidara: 6
Jack Colback: 5
Vurnon Anita: 5
Moussa Sissoko: 6
Gabriel Obertan: 6
Georginio Wijnaldum: 5
Papiss Cisse: 5
Subs
Steven Taylor: 6
Rolando Aarons: 5
Aleksandar Mitrovic: 5
STAR PLAYER
Jefferson Montero: Started where he had left off at Chelsea, namely terrorising full-backs
Newcastle right-back Daryl Janmaat saw red before half-time after stopping the Ecuadorian illegally for the third time and no alternative marker could get close to him
Hit the post from 20 yards and capped another impressive show by setting up Andre Ayew's header which made it 2-0.
MOMENT OF THE MATCH
Andre Ayew: Has needed no time to settle in English football since his summer move from Marseille
The Ghanaian showed wonderful composure to score on his debut against Chelsea last week after a delicious drag-back and here he revealed his aerial ability
Excellent power and direction from Montero cross saw off Newcastle early in the second half.
VIEW FROM THE BENCH
This was the first time Garry Monk and Steve McClaren had crossed managerial swords and it was the younger man who was smiling as his side dominated almost from start to finish
Monk has his cut work out for Swansea to end higher than last season's eighth-placed finish but the early signs are good with plenty of fluid football in a disciplined team framework
McClaren's plan to stifle Swansea were undone by an early Gomis goal and Janmaat's red card - and there was no way back after the latter.
MOAN OF THE MATCH
McClaren must have wondered what his right-back Janmaat was doing to get sent off close to half-time
Facing Swansea flier Montero is not the easiest task in the Premier League and the Dutchman was booked after clattering into him twice early on
But what he was thinking tugging back Montero little more than 30 yards from Swansea's goal, well, heaven only knows but it probably brought a good moan from McClaren at half-time.
WHO'S UP NEXT
Sunderland v Swansea (Premier League, Saturday August 22)
Manchester United v Newcastle (Premier League, Saturday August 22)
Source : PA
Source: PA