The first half was a largely forgettable affair with referee Llyr Ap Geraint-Roberts dishing out 15 penalties. Both sides lost their No 8s with the Dragons' Ieuan Jones and the Ospreys' Morgan Allen receiving yellow cards.
Jones and opposite number Matthew Morgan traded three penalties apiece as the score was locked 9-9 at the break.
Despite the Opsreys having the better of the exchanges in trying conditions, Jones nudged the home side in front with his fourth success.
Thomas barged over from close range but Jones ensured the Dragons stopped the rot to win their first game in six attempts.