A try from Alec Copper and a conversion and penalty from Matthew O'Brien and Sam Bevan respectively, handed the hosts a 10-0 lead inside a dozen minutes.
After that, however, the Scarlets gained the upper hand and following a first half try from Kirby Myhill, they duly completed a second victory of the new season.
After Bevan had kicked a penalty two minutes into the second half and though the Dragons were combative throughout, the visitors won the day thanks to a brace of tries from full back George North and as many conversions from Luke Treharne.
Dragons team manager Billy Vyvian said; "Disappointed, of course, but delighted to have played our part in a very strong game of rugby.
"We had our opportunities to win the game, but we lacked that finishing touch. Still, in terms of our development, it was encouraging."