But with just two minutes left to play Davies was helped off the pitch after turning on his ankle.
The Scarlets dominated the opening half at Rodney Parade and tries by George North and Matthew Rees either side of Davies' effort saw the West Walians race into a 24-3 lead.
The Dragons had started the game brightly and Matthew Jones opened the scoring after seven minutes with a well-struck penalty.
But the Scarlets hit back almost instantly with a fine try. After an initial break by Wales outside-half Stephen Jones, the ball went through several phases and Jones put Davies through a gap. Davies unselfishly drew Will Harries and put North in for his fourth try of the season.
Jones added the extras and soon put the Scarlets further ahead with a penalty to give the West Walians a 10-3 lead.
The Scarlets were beginning to dominate affairs with an astute kicking game while the Dragons struggled to create any go-forward.
But the Dragons were given an opportunity to close the gap after just over 20 minutes but Jones' penalty drifted agonisingly wide of the up-rights.
Moments later Davies struck with a brilliant individual try. During a scrappy period of play, the Scarlets appeared to be going nowhere, but Davies stepped inside Rob Sidoli and showed tremendous pace to beat both Gavin Thomas and Matthew Jones. Will Harries and Adam Hughes' tackle could do little to stop the Welsh centre from crashing over for his sixth try in five games.
Just minutes later the Scarlets extended their lead further and punished the Dragons for their inability to secure their own lineout ball. Decan Manu drove into the Dragons defence, aided by Matthew Rees, but the Men of Gwent were caught off-guard when Manu slipped the ball to Rees to score. Jones again converted to give the visitors a 24-3 half time advantage.
The Dragons began the second half the way they started the first and Matthew Jones chipped away at the Scarlets' lead with a second penalty to bring the score to 24-6.
Unlike the first half, the Dragons continued their pressure and added a third penalty to bring the score to 24-9.
The Scarlets' indiscipline was beginning to show and after 50 minutes they saw their third man sin-binned when Rees was shown yellow.
Will Harries punished the Llanelli-based region for their indiscipline with a well-taken try and the hosts almost added a second after breaking out of their half but Tom Riley failed to find Harries out wide.
It took the Scarlets until the 74th minute to get back on the scoreboard with another Stephen Jones penalty and but it was all they needed to secure a third win of the season and a third in a row at Rodney Parade.
Newport Gwent Dragons
Will Harries; Adam Hughes, Tom Riley, Rhodri-Gomer Davies, Aled Brew; Matthew Jones, Wayne Evans; Phil Price, Tom Willis (capt), Ben Castle, Luke Charteris, Robert Sidoli, Danny Lydiate, Robin Sowden-Taylor, Gavin Thomas
Replacements: Lloyd Burns, Nigel Hall, Aaron Coundley, Adam Jones, Andrew Coombs, Shaun Connor, Nicky Griffiths, Tom Cheeseman
Andy Fenby; George North, Regan King, Jonathan Davies, Sean Lamont; Stephen Jones, Tavis Knoyle; Iestyn Thomas, Matthew Rees, Deacon Manu, Lou Reed, Vernon Cooper, Josh Turnbull, Johnathan Edwards, David Lyons
Replacements: Ken Owens, Phil John, Rhys M. Thomas, Dom Day, Jonny Fa'amatuainu, Martin Roberts, Gareth Maule, Rhys Priestland
Newport Gwent Dragons - Tries: Will Harries Pens: M Jones 3
Yellow card: R Sowden-Taylor
Scarlets - Tries: G North, J Davies, M Rees Pens: S Jones 2 Cons: S Jones 3.
Yellow cards: J Davies, M Rees, T Knoyle