The hosts had made a hot start in freezing conditions and led 10-0 thanks to a try from flanker Sean Morrell and five points from the boot of fly-half Florin Vlaicu.
But the Welsh visitors roared back as Ashley Smith, Rhys Thomas, James Thomas and Andrew Coombs (2) all crossed for tries, with Rhys Jones and Jason Tovey booting the rest of their points.
It leaves the Dragons second in the pool, behind Newcastle who have a 100 per cent record from their three games.
But a bonus point win had not looked likely at Ghencea Stadium, which was hosting the fixture after the pitch at Stadium Arcul was left waterlogged by melting snow.
Vlaicu’s boot had given the home side an early lead, and they extended that advantage when a series of strong forward carried ended with Morrell charging over for a converted try.
But the Dragons began to take control as they adjusted to the testing conditions, with Jones and Tovey slotting a penalty apiece to bring them back to within four points.
They landed a crucial blow just before half-time as a patient build-up ended up working Smith over for their first try, converted by Jones for a 13-10 lead.
Two tries early in the second half effectively killed the game off as two well controlled driving mauls allowed Rhys Thomas and James Thomas to dot down and out the away side 10 points clear.
The scores kept on coming as a Tovey interception paved the way for Coombs’ first, with the Wales international rounding off the scoring with his second just after the hour mark.
