The Scarlets took the lead with a penalty try in the first half before the home side hit back with tries from Ashley Smith and Mustoe to earn their first Magners League double over Phil Davies's men.
A jubilant Mustoe said: "It's a great win for us and it was much needed because we haven't been getting the results. It's massive because the pressure has been building on us over recent weeks and we were playing for our careers."
British Lions and Wales fly half Stephen Jones missed the Rodney Parade derby clash with a bruised thumb. But his deputy Rhys Priestland confidently opened the scoring with a close-range penalty on seven minutes.
The Scarlets looked threatening in attack but they also proved they had the muscle up-front. And their forward dominance paid off with a penalty try on 33 minutes as the Dragons collapsed a scrum on their own try-line. Referee James Jones had no hesitation and signalled under the posts and Priestland converted to hand the Scarlets a 10-point advantage.
Jones sent Scarlets flanker James Bater to the sin bin on the stroke of half time for killing the ball and Ceri Sweeney cut the lead to seven points.
Sweeney then turned provider by offloading to centre Smith who touched down under the posts on 52 minutes. The Dragons fly half and Priestland exchanged penalties before Mustoe brought some festive cheer to the home faithful with his late but clinical finish.