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Dawson went around the blindside of the lineout to touch down under the posts and Australian full-back Clinton Schifcofske converted.
Dragons' James Arlidge opened the visitors' account with a penalty after 19 minutes, awarded following an infringement by Ulster at the lineout.
Arlidge cut Ulster's lead to a point after 30 minutes when he slotted over his second penalty from just inside the home side's half.
Ulster could not turn their pressure into points and allowed the Dragons to take a 9-7 lead at the interval when Arlidge kicked a third successful penalty in first half stoppage time.
Former Ulster number eight Grant Webb was unlucky not to extend his new club's lead when he broke out of defence and went nearly the full length of the pitch, and only a desperate tackle from Timoci Nagusa stopped him from scoring a try.
The Dragons had left-winger Richard Fussell sin-binned on 50 minutes and Ulster should have capitalised when Andrew Trimble broke out of his own half but was forced into touch near the try-line.
Against the run of play the Dragons broke out of defence and increased their lead with a try from centre Ashley Smith.
A kick from Arlidge caused confusion in the Ulster defence as they let it bounce.
Arlidge gathered and was caught short of the line, but as he tried to offload in the tackle an Ulster hand knocked the ball back over the line. That gave Smith the chance to pounce and score with Arlidge converting.
Replacement Ian Humphreys crossed for Ulster with four minutes left after a sustained period of pressure which saw Darren Cave cut through the defence, allowing the former Leicester Tiger to go in from close range and Schifcofske to set up an exciting finale, but the home side had to settle for a third consecutive bonus point.