Swansea, who won the Capital One Cup on Sunday, started brightly. A wonderful dribble from Pablo Hernandez, which saw him leave Mathieu Debuchy and Yohan Cabaye trailing in his wake, ended in a chance for Michu which the Spaniard headed over.
In the ninth minute Michu 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.
The game became increasingly niggly and Davies was fortunate not to be carded after a late challenge on Jonas Gutierrez, before Cheick Tiote and Cabaye were booked in quick succession for fouls on Nathan Dyer.
Swansea carved the visitors open with seconds of the restart - only for goalkeeper Elliot to superbly deny Angel Rangel from Hernandez's pass.
Newcastle spurned their two best chances of the afternoon a few minutes later. Papiss Cisse volleyed wildly over, before Yoan Gouffran made a hash of finishing from Moussa 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 to save from Sissoko.
However, after Swansea substitute Ki Sung-yueng had been denied by Elliot, the hosts won it when Moore prodded home - with the help of a deflection off Cabaye - after Davide Santon had failed to clear Wayne Routledge's cross.