The Scarlets' first league victory since October has closed the gap on the seventh-placed Blues to just two points, but the result was tempered by a shoulder injury picked up by out-half Stephen Jones.
The Wales and Lions star was forced off early in the second half, complaining of soreness on the A/C joint of his right shoulder. His withdrawal came just minutes after he had kicked the visitors back in front at 11-9.
The regional rivals were short of a number of frontline players due to injuries. The home side were missing seven first choice men including Welsh international Dan Lydiate, while the Scarlets were shorn of eight players.
The Dragons had suffered a second half collapse against the Blues last Sunday and were eager to get back on the winning trail, especially following the announcement that Colin Charvis has taken over as their forwards coach until the end of the season.
Paul Turner's charges, winners of their last eight games in Newport, made a sluggish enough start, however, and the second-from-bottom Scarlets enjoyed most of the territory in the opening quarter.
The Llanelli-based outfit, who has teenage scrum half Gareth Davies making his first start, notched the only try of the match, although they should have added another two.
Full-back Rhys Priestland was put over for that touchdown after centre Jonathan Davies, who used his power and pace to cause problems in midfield, went past opposite number Matthew Watkins and put Priestland in the clear.
Watkins had earlier put in a superb tackle to knock the ball from the grasp of Scarlets winger Andy Fenby who looked poised to score as he crossed the try-line.
The home side relied on the boot of experienced out-half Shaun Connor who kicked an early drop goal from in front of the Scarlets' posts before landing two penalties, Jones adding one of his own for the visitors.
Back rowers Gavin Thomas and Lewis Evans did a lot of good work at the breakdown for the Dragons, with Connor's penalty successes moving the hosts into a 9-8 lead.
Jones booted another penalty after the break but missed a 35-metre effort in front of the posts and was replaced by Dan Evans immediately to have his shoulder iced on the bench.
As both Turner and Scarlets boss Nigel Davies rang the changes, the Dragons held the upper hand in terms of territory, although they could not find a score, with Connor missing a long range penalty just after the hour mark.
And when Priestland knocked over a tremendous touchline penalty with eight minutes left, the home side needed a converted try as they looked to steal a win that would have moved them up to third in the table.
They tried desperately and battered away in the Scarlets' 22 for an extraordinary 35 phases of play, with back row replacement Grant Webb and prop Pat Palmer unable to ground the ball as the Dragons got within inches of scoring a try in the final play of the match.
The Scarlets and man-of-the-match Priestland, who had moved to out-half to cover for Jones, did enough to hold on and take an important four points in the scheme of their season.
Notably, it was a fifth win in their last six trips to Rodney Parade and very much founded on a gritty and determined defensive display, particularly during that late Dragons' siege.
Speaking afterwards, Scarlets coach Nigel Davies said: "We were desperate for the win and that was phenomenal defence, the boys were very focused.
"We showed our determination and as a group we will take a lot out of that for the rest of the season. It was important that we showed we can go away from home and win."
Commenting on Stephen Jones' injury, he added: "He took a bit of a bang on his A/C joint. I don't think it's anything too serious. We'll monitor him over the next couple of days."
Dragons supremo Paul Turner reflected the disappointment of his dressing room when saying: "It was very disappointing that we didn't score a try (at the end0, but we just made too many errors.
"I thought that we should have had a couple more penalties. When they took Stephen Jones off they shut up shop.
"I can't fault the effort, but at the top level you have to cut down the errors. You have to win your home games."
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