Rhys Jones got the visitors off to the perfect start at Bridgend's Brewery Field by slotting a penalty before outside centre Ross Wardle bundled over in the eighth minute as the Dragons quickly opened up a 10-0 lead.
That lead was extended further when back-rower Lewis Evans scored two more tries for Newport.
Jones added another three points as the Dragons threatened to finish the game as a contest before the half-time whistle.
Such was the away side's dominance that they were able to wrap a try-scoring bonus point on the half-hour mark through prop Cory Hill as the Ospreys struggled to find a response. The hosts' cause was further hindered by Rory Thornton's yellow card on 29 minutes.
As the half-time whistle approached, scrum-half Jonathan Evans crossed over the whitewash for the Dragons as they reached the interval with a 41-0 advantage.
Cory Hill crossed for another score to extend the Dragons' lead to 48 points, but the Ospreys produced a second-half fightback with tries from Hanno Dirksen, Dafydd Howells and replacement Cai Griffiths, giving the hosts faint hope of an unlikely victory. The Ospreys were then awarded a penalty try on 66 minutes as the gap closed to 51-28.
However, the Dragons were able to see out the final minutes and pick their first LV= Cup win of the season.