Early tries from Josh Lewis and Jared Rosser left Wales’ Capital Region with an uphill challenge in Newport but after slowly building into the encounter they burst into life.
Owen Lane squeezed in the corner following a dominant period to give his side hope before the break and Anscombe kept the scoreboard ticking.
After cutting the Dragons advantage to just four points, the Blues struck with two tries in five minutes.
Kristian Dacey was driven over from an attacking lineout and Lane grabbed a second impressively following an electric Gareth Anscombe break.
The victory was the Blues seventh in a row over the Dragons in the competition as the Dragons slumped to a 22nd consecutive Welsh derby loss.
The Blues displayed some early enterprise in the encounter with Willis Halaholo, Rey Lee-Lo and Matthew Morgan making good yards but the majority of play was in their own half.
Refereeing decisions allowed the home side to control the territory but Josh Lewis pushed a long-range effort well wide.
The Dragons had further opportunities when and Ian Davies penalty allowed Lewis to push the hosts into the corner but a huge hit from Nick Williams halted Brandon Nansen before Cory Hill was held up over the line.
The Dragons were edged back from the ensuing scrum and the Blues had the opportunity to clear after Josh Turnbull stole possession.
However, they were penalised once again at the breakdown and the Dragons went to the corner. Elliot Dee was half agonisingly short from a set move at the lineout before the ball was worked left.
It looked as though Jordan Williams had wasted the overlap as he was caught but he offloaded off the floor to Lewis on the loop and he squeezed in the corner.
Josh Lewis missed the conversion, despite having two attempts following an early charge, but the Dragons didn’t have to wait long to stretch their lead.
The opportunity arose in fortuitous circumstances as the Dragons latched onto their own charged down kick. A couple of phases later Adam Warren sliced through the Blues midfield and found Jared Rosser in support.
Rosser finished with the effort converted by Jordan Williams, leaving the shell-shocked Blues 12-0 following an opening quarter dominated by the hosts.
A thunderous Anscombe kick, which was dropped by Williams, allowed the Blues the opportunity to claw their way back into the game as Lane surged forward.
They spent the remainder of the half camped in Dragons territory with a Willis Halaholo try ruled out by the TMO after replays showed Warren had got a hand to Lloyd Williams’ intelligent infield grubber first.
The Blues scrum was proving to be a major weapon and allowed them to remain in the red zone as the Dragons defended desperately.
Harries went close on a succession of occasions and Lee-Lo was also unfortunate before the breakdown eventually came.
It appeared as though the fired up hosts had snaffled out another Blues opportunity when Lane was caught five-metres short but he did remarkably well to bounce out of contact before driving through two defenders to squeeze in the corner by a hair.
Anscombe was unable to convert but the try had given them hope and they dominated the remainder of the half with the scores remaining 12-5.
Wales’ Capital Region made a promising start to the second half with the scrum continuing to cause the hosts problems and Anscombe slotting a lonf-range effort to cut the deficit to four.
Harries was enjoying plenty of opportunities and he kicked ahead and outstripped the Dragons defence to collect on the left. He was just about caught and at the ensuing breakdown the ball was knocked on.
With both teams battling for the territorial ascent, kicking duals broke out with the Dragons desperate to keep the Blues pinned in their own half.
Another long-range effort from Anscombe brought the Blues within one-point of their East Wales rivals but it was quickly chalked off by Jordan Williams.
The Blues took the lead for the first time shortly before the hour mark with a penalty kicked to the corner and a rampaging driving lineout sending Dacey over.
Anscombe’s conversion sailed narrowly wide but he was the architect of a stunning team score five minutes later.
The fly-half took a high ball under huge pressure before embarking on a high risk, high reward counter-attack.
He beat two defenders as he raced to the corner before being scragged by Rosser. But possession was maintained and worked right with a cut out Jarrod Evans pass putting Lee-Lo in space.
The Samoan centre drew one defender before feeding Lane, who burst through one tackle to dive acrobatically into the corner.
Mulvihill's side now head into back-to-back European fixtures against Lyon and Glasgow Warriors, as they make a long-awaited return to the Heineken Champions Cup.