And Reivers only had themselves to blame, missing a series of clear-cut chances to put crucial points on the board when it mattered.
Rain during the build up to kick-off left conditions extremely greasy and both sides found it almost impossible to string together decent passes.
Reivers were handed the first scoring opportunity with a penalty for handling in a ruck, but Calum MacRae sent his 42 metre effort wide.
Dragons made the most of the let-off by manufacturing a touchdown in the 12th minute. They pinned back the home markers with a pick-and-drive sequence by the forwards, which paved the way for prop Adam Black to plunge over the line.
Ceri Sweeney added the extra points to give the Dragons a useful platform.
Borders shook off the setback to mount a series of powerful raids and they claimed a clever try of their own in the 32nd minute.
Stand-off Steve Jones’ chip sparked hesitation in the Newport defence. They allowed the ball to bounce — and it popped into the arms of Nick de Luca, who cruised to the target.
Sweeney snatched back the momentum for the Dragons with a penalty for ball-killing, however there was still time before the interval for MacRae to respond with an identical strike.
Irish international referee Simon McDowell was forced out of the game at half-time with a groin strain and was replaced by senior touch-judge Andy Ireland.
Reivers looked more focused after the restart, but they could not turn pressure into scores.
MacRae was off the mark with his next penalty, then hooker Ross Ford spilled the ball close to the line when he looked capable of going over.
There was even more frustration as Opeta Palepoi was brought down just a pace short of the target area.
And at the other end, there was a let-off for the home side when a Sweeney penalty crashed against the upright and stayed out.
But he made no mistake with the conversion of Kevin Morgan’s try and his second penalty saw off Reivers.
Borders scorers: Tries: De Luca. Pens: MacRae.
Newport-Gwent D’gons: Tries: Black, Morgan. Cons: Sweeney 2. Pens: Sweeney 2.
BORDERS: MacRae, Danielli, De Luca, Law, Robertson, Jones, McKearchar, Kerr, Ford, Douglas, Palepoi, Grimes, Brown, Miller, Beattie.
Replacements: Townsend for MacRae (58), Chrystie for McKearchar (61), Kalman for Douglas (52), Dalziel for Miller (70), Gray for Beattie (58).
NEWPORT-GWENT D’GONS: Morgan, Wyatt, Emerick, Smith, Fussell, Sweeney, Walker, Black, Daly, Thomas, Gough, Sidoli, Bearman, Ringer, Charvis.
Replacements: Hercus for Smith (61), Jones for Daly (37), Harrison for Thomas (42), Charteris for Sidoli (63), Fitisemanu for Ringer (77).
Referee: S McDowell (Ireland).