Gomis opened the scoring in the ninth minute with a composed finish at the Liberty Stadium.
Jonjo Shelvey threaded a through ball for Gomis to race onto. Newcastle waited in vain for an offside flag as Gomis rounded Newcastle goalkeeper Tim Krul.
Ayew, who was also on target in last weekend's 2-2 draw at Chelsea, scored with a powerful header in the 52nd after Newcastle fullback Daryl Janmaat was sent off for a second yellow card.
Janmaat was booked after clattering into Jefferson Montero twice and he went off in the 41st when he pulled the Ecuadorian back to stop him launching an attack.
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