Jordan Thompson missed numerous first-half chances for the hosts, while Brian Gilmour's effort amounted to nothing.
Paddy Boyle provided Moore with the perfect cross on the stroke of half-time, though, and the Motherwell loanee tucked home his ninth of the season.
Ayr stopper Greg Fleming denied Jamie Bain a leveller.
Craig Moore claimed a first-half winner as Ayr United strengthened their hold on third place in League One with a narrow victory over Airdrieonians.