Visitors Edinburgh made the fatal error of allowing the hosts to get into their groove and will need to assert themselves quicker on their upcoming travels to Connacht and Ulster.
Michael Bradley's men have now not won in the RaboDirect PRO12 since December and only once away from Murrayfield all season.
But they made a good start when fly-half Phil Godman opened the scoring in the third minute but Tovey replied in the 16th minute before taking the Dragons into the lead three minutes later with his second.
It was then Edinburgh's turn to have a man binned with lock Sean Cox in trouble for illegal rucking. That set the Dragons up for their flurry of points and it was Bedford who started it.
The scrum-half showed his forwards how to barge over from a yard out in the 24th minute after hooker Steve Jones went mightily close, Tovey calmly adding the extras.
Then the Dragons backs had a go and after a strong break through the middle by centre Anitele'a Tuilagi, Bedford passed to full-back Martyn Thomas who slipped the ball to Brew for a try in his 100th game for the region.
Brew is leaving at the end of the season to go to an as-yet unannounced European club.
But it will perhaps be Tovey that the fans will miss most, particularly as it will be a Cardiff Blues jersey on his back from next season.
With the score at 18-3 Edinburgh again subdued the Dragons for the first 20 minutes of the second half but this time made the running themselves and got a try back in the shape of Grant Gilchrist. Replacement Gregor Hunter converting.
It was a true forwards try, coming after a five-metre scrum and from the follow-up ruck, Cox acted as scrum-half and fed second-row partner Gilchrist who muscled over.
A Tovey penalty just before Edinburgh's score kept the Dragons comfortable but when Tim Visser hacked a loose ball out of the Gunners' 22 the 11-point gap should have been cut.
Visser re-collected his kick and the ball found its way to centre Matt Scott but with no Dragons player putting any true pressure on him he attempted an out-the-back pass to winger Sep Visser who knocked-on.
That proved to be the Gunners' last chance to get back into the game and the Dragons now go to Italy in a bottom-of-the-table clash at the Stadio Zaffenella eager to build on this win.
|Newport Gwent Dragons Score Card|
|Edinburgh Rugby Score Card|