The Irish province had come into the game looking to bounce back from defeat in Wales last Sunday, going down 32-14 to the Ospreys at the Liberty Stadium, to lose their unbeaten record this season.
But they were unable to handle the quick turnaround as the Dragons recorded their second home win after beating Aironi in their first outing at Rodney Parade.
Rookie outside half Paddy Jackson had given Ulster an early lead but in a sign of what was to come, indiscipline allowed the Dragons to come back into it through two Jason Tovey penalties.
Worse was to come as centre Nevin Spence was sent to the sin-bin for a high tackle on the new-boy Chavhanga who had already shown his pace on one mazy run.
The home side took full advantage of the extra man as centre Tom Riley burst through the gap left by the missing
Spence and escaped the clutches of Luke Marshall to cross on 25 minutes and put Dragons 13-3 up after Tovey converted.
Jackson and Tovey exchanged penalties before Tovey added yet another kick just before the break as Ulster were again penalised for offside to make it 19-6 at the break.
The second half began as the first ended with Tovey adding another three points to stretch the lead further for the Dragons.
Jackson countered that with three more points of his own as Brian McLaughlin's side finally appeared to come to terms with referee Neil Paterson's interpretations.
Ulster winger Ian Whitten thought he had got the visitors back into it with a try in the corner with 12 minutes to go, but a fantastic covering tackle from Jason Tovey forced him into touch just short of the line.
And despite a wonderful late pickup from Jackson, Ulster could not find another score as the young pivot was tackled, McLaughlin's going home empty-handed for the second time in a week in the Principality.
|Newport Gwent Dragons Score Card|
|Ulster Rugby Score Card|