Rhys Priestland, who deputised for back injury victim Stephen Jones, did all the scoring for the hosts with six successful kicks, ably assisted by man-of-the-match flanker Dafydd Jones.
Jones will be hopeful of making the Wales squad for the November internationals, which will be announced on Sunday, but injuries to Stephen Jones and Dragons duo Luke Charteris and scrum half Wayne Evans, who both have neck complaints, will concern the Welsh management.
Scarlets winger Mark Jones also limped off with a knock, although the news about the quartet is largely positive.
There were few positives for Dragons coach Paul Turner to take from the game, and he admitted as much afterwards.
"We just weren't at the races, a penalty count of 19-7, our set piece was pretty poor and we missed Luke (Charteris), Tom (Willis), who went off early, and Joe Bearman," said Turner.
"But, at the end of day, we were just not there and were second best in every department. The Scarlets got the momentum and six penalties to one was probably just desserts for them."
The Dragons travelled having won three of their first five league games, but were chasing only their second ever win in Llanelli since the formation of the regions in 2003.
And they never looked like returning to Gwent with victory after a lacklustre showing in a derby clash that ultimately failed to catch fire.
The conditions, particularly during a rain-hit second half, gave these Welsh rivals little leeway in terms of their style of play.
Number 10 Priestland broke the deadlock with a 13th minute penalty before kicking another after 24 minutes, either side of James Arlidge's sole success for the Dragons.
Moments after Priestland's second penalty, the Dragons nearly scored the game's first try. Sean Lamont missed a tackle and the ball squirmed free over the line with Richard Fussell and Martyn Thomas in pursuit before it was grounded amid some desperate Scarlets defence.
Both sides were struggling to make any impact as the first period drew to a close - although the Dragons, who lost the services of returning captain Tom Willis early on, again nearly snatched the lead as half-time approached.
Full-back Thomas found himself in space and kicked through for Ashley Smith, who gave chase before the ball trickled into touch just metres from the Scarlets line.
The game might have ignited four minutes after the restart when Scarlets prop Iestyn Thomas burst through, before offloading to David Lyons, but with the big number 8 nearing the line, referee Peter Allan pulled play back for crossing.
Priestland gave the hosts some breathing space with two further penalties from distance in the 50th and 54th minutes.
Another Priestland strike strengthened that grip further before Dragons winger Aled Brew was sin-binned with 13 minutes remaining.
That was the final nail in the Dragons coffin and Priestland coolly sealed the win with his sixth penalty from as many attempts.
|Scarlets Score Card|
|Newport Gwent Dragons Score Card|