Swansea, who have spent the last six days celebrating their Capital One Cup success, looked set to be frustrated after missing a host of excellent chances.
But Moore got his first Premier League goal of the season from close range five minutes from time as he prodded home, with the help of a deflection off Yohan Cabaye, after Davide Santon had made a mess of clearing a Wayne Routledge cross.
Defeat continues the Magpies' miserable away record - they have won just once on the road in the league - and leaves Alan Pardew's side still glancing nervously over their shoulders.
The trio of Pablo Hernandez, Routledge and Nathan Dyer caused chaos all afternoon with their pace, precision and movement.
Michu was then allowed to drift around the back from a deep free-kick and his flick past the advancing Rob Elliot left Ashley Williams with a simple finish, but the Wales captain scuffed his effort and Steven Taylor cleared off the line.
Mathieu Debuchy had a slice of fortune when the ball struck his hand after an aerial challenge with Ben Davies, but referee Craig Pawson, on his Premier League debut, waved away Swansea appeals.
Swansea were clearly being troubled by Newcastle's direct approach, but they carved the visitors open with seconds of the restart - only for Elliot to superbly deny Angel Rangel from Hernandez's lovely pass.
Newcastle spurned their two best chances of the afternoon a few minutes later as they took a hold on proceedings.
Firstly Papiss Cisse, who scored twice in Newcastle's win here last season, wildly volleyed over the bar, before Yoan Gouffran made a hash of finishing from Moussa Sissoko's pull-back.
Michel Vorm tipped a Yohan Cabaye shot onto his bar as Newcastle ratcheted up the pressure, and the Dutchman was again at full stretch in the 73rd minute to tip a Sissoko drive around the post.
However, Vorm's opposite number Elliot had little chance of keeping out the winner as Cabaye's attempted challenge took Moore's strike beyond him and into the corner of the net.