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 Barclays 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.
Newcastle spurned chances of their own after having the better of the second half, with Papiss Cisse, Moussa Sissoko and Yoan Gouffran proving particularly wasteful.
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.
Swansea, meanwhile, reached the 40-point mark to cap the most memorable of weeks for the south Wales club.
With the League Cup proudly on display before kick-off, Michael Laudrup's side had started the match in the manner in which they had dissected Bradford at Wembley last weekend.
The trio of Pablo Hernandez, Routledge and Nathan Dyer causing chaos once again with their pace, precision and movement.
A wonderful dribble from Hernandez, which saw him leave Mathieu Debuchy and Cabaye trailing in his wake, ended in a chance for Michu, but Swansea's talisman could only head over.
Michu was again involved as the hosts spurned a great chance in the ninth minute.
The Spaniard was 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.
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.
Pardew had seethed at his side' sluggish opening, but the Magpies slowly but surely established a foothold in the game, harrying and physically challenging Swansea's ability to play their passing patterns.
The game became increasingly niggly and Ben Davies was fortunate not to be carded after a late challenge on Jonas Gutierrez, much to Pardew's annoyance.
Temperatures rose further as Cheick Tiote and Cabaye were booked in quick succession for fouls on Dyer.
Leon Britton and Papiss Cisse then exchanged words as the half-time whistle was blown.
Pardew unwisely opted to add his voice to the disagreement as the Swansea midfielder walked past him on his way down the tunnel, although players from both sides quickly stepped in.
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 Cisse, who scored twice in Newcastle's win here last season, wildly volleyed over the bar, before Gouffran made a hash of finishing from Sissoko's pull-back.
Michel Vorm tipped a 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.
Swansea went close when Hernandez's disguised pass put substitute Ki Sung-yueng one-on-one with Elliot, but the keeper was again equal to the task.
However, he 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.