Centre Steve Winn spearheaded the success by grabbing a brace as the Welsh side ran in a total of five tries, four of which were converted by Percy Montgomery, who also notched three penalties.
Their Scottish visitors showed signs that they could perhaps score their first away victory in the league when they raced to an early nine point lead thanks to three penalties from Charlie Hore.
Errors littered the Dragons’ play as they tried to get their attacking options going and it was left to Montgomery to put them on the board.
Another penalty from the South African reduced the deficit but Borders failed to snuff out their recovery when Simon Danielli and then Stuart Moffat failed to cross after a defence-splitting run by the former.
A fourth Hore penalty nudged the visitors further in front but shortly after a superb burst and offload by prop Rhys Thomas gave Jamie Ringer a clear run to score.
Montgomery wiped out the Borders´ lead to put the Welshmen ahead 13-12.
When the visitors were reduced to 14 men when Wayne McEntee was sin-binned, the Dragons stepped up their pressure and should have capitalised but Rhys Oakley spilled the ball when he should have scored.
However Winn soon grabbed his first five-pointer.
The Borders were up against it again when centre Callum MacRae yellow-carded but they battled bravely.
But they could not stop Winn grabbing his second when he brushed aside the joint challenge of John Weston and Danielli.
The bonus point was secured when Gareth Wyatt grabbed the fourth try and the scoring was completed when Rhys Thomas powered over from a lineout.
Final score: Dragons 42 (Ringer, Wyatt, Thomas, Winn 2, try, Montgomery 4 conv, 3 pen), The Borders 15 (Hore 5 pen)