The Dragons recovered from a sluggish start to score five tries against the Celtic League's bottom side.
Dragons (20) 42
Tries: J Ringer, S Winn (2), G Wyatt, R Thomas
Cons: P Montgomery (4)
Pens: P Montgomery (3)
Borders (12) 15
Pens: C Hore (5)
Fly-half Charlie Hore gave the Borders an early 12-0 lead with four penalties, but one more second-half penalty was all the Scots could muster at Newport.
Jamie Ringer's try started the Dragons comeback before Steve Winn crossed twice either side of half-time.
Gareth Wyatt and Rhys Thomas added late tries, while full-back Percy Montgomery added 17 points with the boot.
The South African also denied the visitors' their only real try-scoring opportunity when winger Stuart Moffatt threatened in the left corner.
That scare near the half-hour mark seemed to focus Dragons minds as they finally found some attacking purpose.
Quick turn-over ball provided the perfect opportunity to spread the ball wide where prop Thomas made the break as the Borders defence stood off.
He was finally brought down by two covering defenders but not before giving a perfectly timed pass to Ringer on his right shoulder who had a clear run to score.
The Dragons were now enjoying a purple patch and more attacking pressure forced flanker Wayne McEntee to kill the ball near his own tryline.
That offence earned him a yellow card, and less than minute later the Dragons crossed for their second try as a simple midfield move from a scrum allowed centre Winn to burst through some weak tackling.
Montgomery kicked his second conversion to give the Dragons a healthy 20-12 half-time lead.
More Borders indiscipline brought a yellow card for centre Callum MacRae and gave Montgomery a simple opportunity to add another three points.
A fifth Hore penalty cut the gap to eight points at 23-15, but two minutes later Winn scored the converted try which sealed the Dragons' win with just over 10 minutes to play.
The ball passed through the hands quickly to create space down the left wing, but the Borders defence again looked rather tame as Winn powered his way over.
There was little the Borders could have done to prevent the home side's next score, though, as the Dragons exploited an over-stretched defence to secure a bonus-point try.
A Scottish hand barely touched a Dragons player as the backs span the ball wide with purpose and set winger Wyatt free for the best try of the match.
There was still time for prop Thomas to benefit from a free ball at the tail of a Borders line-out and charge his way over from five metres for another converted try.
Dragons: P Montgomery; G Wyatt, S Tuipulotu, S Winn, B Breeze; C Warlow, G Baber; A Black, J Richards, R Thomas, I Gough, P Sidoli, J Ringer, J Forster (capt), R Oakley.
Replacements: G Robinson, R Snow, C Anthony, L Charteris, M Veater, J Ireland, J Bryant.
Borders: S Moffat; S Danielli, C MacRae, G Law, N Walker; C Hore, J Weston; T McGee, R Ford, B Douglas (capt), M Blair, C Stewart, J Dalziel, W McEntee, K Brown.
Replacements: G Cross, W Kay, A Turnbull, G Morton, Wilson, A Dickson, J Pattison.
Referee: Donal Courtney (Ireland).