Nigel Davies' men began their campaign with victory at Ulster and produced a storming second-half comeback to sink the Dragons at Rodney Parade.
The Dragons looked set for their first win of the season following first-half tries from Richard Fussell, Tom Willis and Lewis Evans.
They led 25-6 at the break but the Scarlets rallied through tries from Martin Roberts and Nathan Brew before Jones booted home the late winner.
Wales coach Warren Gatland was in attendance as a host of his Grand Slam winners were in action and he will have been impressed with first-choice fly-half Jones' nerve late on.
Jones, who skippered the Scarlets in the absence of fellow Lion Simon Easterby, will be hoping his impressive form continues throughout teh season as he looks to secure his place on a second successive Lions tour.
Tries: Fussell, Willis, Evans
Cons: Arlidge (2)
Pens: Arlidge (2)
Tries: Roberts, Brew
Cons: S Jones
Pens: S Jones (5)
Dragons: P Dollman (S Connor 33); T Selley, R Sidey, M Stcherbina, R Fussell; J Arlidge, W Evans (R Lewis 68); A Black (H Gustafson 58), T Willis (capt), R Thomas (L Harrison 73), A Jones (G Webb 70), L Charteris, J Bearman, L Evans, R Parks
Unused replacements: J Ringer, G Wyatt
Yellow Card: Selley 53
Scarlets: M Stoddart; D Daniel (N Brew 50), R Higgitt, G Evans, D James; S Jones (capt), S Martens (M Roberts 52); I Thomas, M Rees, K Meeuws (D Manu 67), L Reed (S MacLeod 52), V Cooper, D Jones (S Easterby 52), D Lyons, G Thomas
Unused replacements: K Owens, C Thomas.