The Dragons got off to an almost perfect start when they caught the Borders cold with a try inside four minutes.
Wales' former vice captain Michael Owen made ground to set up a ruck before scrum-half Gareth Baber swung the ball wide for full-back Gareth Wyatt to put in flying wing Gareth Chapman from 20 metres.
Outside-half Aled Thomas missed the conversion but, after a few moments of Borders' pressure, the Dragons' stand-off made amends with a thumping drop goal from 40 metres.
The Borders only managed to get close to scoring once when, early in the match, they had an attacking line-out within metres of the home line.
But, despite losing men like captain Kevin Morgan, number eight Colin Charvis and number 10 Ceri Sweeney, told to rest as part of Wales' RBS Six Nations build-up, the Dragons put their stamp on the match.
And it was not long before the Dragons made their presence felt. This time, it came through the forwards as lock Luke Charteris took a high line-out throw and delivered it to prop Adam Black who was rumbled over the line from five metres.
Thomas added the extra points and the Welshmen had a good platform on which to build the rest of the game.
But the turning point came after the break and it was of the Borders' own making.
At 15-0 down at half-time, at least they were still in with some chance of performing a great comeback but the loss of full-back Calum MacRae for a needless high tackle on flanker Jamie Ringer reduced his team to 14 men.
And, with that, all hope of victory went as the Dragons scored three tries in the space of seven minutes. Again, Owen was at the forefront of matters when he sent Black, standing on the wing, over from five yards.
Within moments, big number eight Nic Fitisemanu finished off a superb movement that saw centre Ashley Smith just caught by the corner flag before the ball was thrust into midfield and Fitisemanu could not be stopped.
Before MacRae could return to the fray, scrum-half Alex Walker sent the whole Borders side the wrong way to dummy at a ruck and waltz in unopposed.
Then two dashing tries down the left by Wyatt and the right by replacement wing Richard Fussell in injury time, both of which were converted by Thomas, sent the Dragons fans home in great heart.
The scorers:
For Newport Dragons:
Tries: Chapman, Black 2, Fitisemanu, Walker, Wyatt, Fussell
Cons: Thomas 5
DG: Thomas
For Border Reivers:
None
Yellow cards: MacRae (Border, 48 Min, High tackle)
The teams:
Dragons: 15 Gareth Wyatt, 14 Gareth Chapman, 13 Paul Emerick, 12 Ashley Smith, 11 Aled Brew, 10 Aled Thomas, 9 Alex Walker, 8 Nic Fitisemanu, 7 Jamie Ringer, 6 Michael Owen (c), 5 Peter Sidoli, 4 Luke Charteris, 3 Jamie Corsi, 2 Ben Daly, 1 Adam Black
Replacements: 16 Richard Wilkes, 17 Gethin Robinson, 18 Andrew Hall, 19 James Harris, 20 Richard Fussell, 21 Phil Dolman, 22 Gareth Baber
Border: 15 Calum MacRae, 14 Simon Danielli, 13 Nick De Luca, 12 Ben MacDougall, 11 Garry Law, 10 Gregor Townsen, 9 Chris Cusiter (c), 8 Scott Forrest, 7 Scott Newlands, 5 Opeta Palepoi, 4 John Dalziel, 3 Bruce Douglas, 2 Ross Ford, 1 Ryan Grant
Replacements: 16 Steve Scott, 17 Ed Kalman, 18 Richie Vernon, 19 Roddy Grant, 20 Rob Chrystie, 21 Steve Jones, 22 Dougie Flockhart
Referee: Alain Rolland (Ireland)
Touch judges: Phil Connett (Wales), Alan Shepperd (Wales)