Second-half tries from wing duo David Lemi and Tom Varndell saw Wasps turn over the Dragons 23-16 at Cardiff City Stadium on Sunday.
The game, switched from a frozen Rodney Parade, never took off as a spectacle.
But Wasps prevailed through scores by Ben Jacobs, Lemi and Varndell despite producing a performance that fell way below their best.
The Dragons, who led 10-0 and then 16-10, managed an early Ashley Smith touchdown and 11 points from fly-half Jason Tovey - a drop-goal, two penalties and conversion - before slipping to their third successive group defeat.
They also had two tries ruled out by video official Iain Ramage and dominated the final quarter, yet Wasps still headed back along the M4 with four crucial points.
Wasps are two points behind pool leaders Toulouse in the race for an automatic quarter-final place, with the French giants yet to visit Adams Park.
The Londoners have now won five successive games in all competitions, yet they will require a considerable improvement to trouble Toulouse, with the French heavyweights looking well on course for a fifth European title.
The Dragons, without a Heineken Cup win since they beat Glasgow more than 12 months ago, tackled double European champions Wasps without injured Wales forwards Luke Charteris and Dan Lydiate.
Wasps, meanwhile, paraded Wales back-row forward Andy Powell at his old Cardiff Blues stomping ground as they targeted an away success.
But it was the Dragons who came out firing, roaring 10 points clear in as many minutes.
Tovey booted a 40-metre penalty to set the ball rolling, and with Wasps struggling to break free from their own 22, slick passing by Tovey and Wales wing Aled Brew sent Smith over unopposed.
Tovey added the extras, and there was no immediate sign of Wasps stirring as they continued to make errors up-front and behind the scrum.
Walder opened the visitors' account with a penalty after his first attempt hit the post, but the Dragons continued to press as they maintained territorial dominance.
Wasps, though, drew level in enterprising fashion, England lock Simon Shaw turning defence into attack through some superb work inside his own 22 before Jacobs and Lemi combined to take play a further 60 metres downfield.
And with the Dragons on the back foot defensively, Jacobs sliced through a gap to score. Walder converted to make it 10-10, and then kicked another penalty, but only after a Tovey penalty and drop-goal restored the home side's advantage.
Wasps, with full-back Mark van Gisbergen enduring a difficult afternoon under the high ball, strung little together in terms of sustained attacking play.
But they had a reliable points source in Walder, who added another penalty just before the break, enabling Wasps to troop off 16-13 adrift.
The Dragons saw prop Nigel Hall carried off with what appeared to be an ankle injury, and matters then deteriorated for the home side when Tovey was hurt after making an abortive attempt to halt a flowing Wasps move that ended with their second try.
Centre Steve Kefu ghosted through the Dragons defence, and Lemi did not require any second invitation from 20 metres out, scoring with his trademark celebratory dive.
Although Walder missed the conversion, Wasps led for the first time as Tovey was helped off to be replaced by Matthew Jones, who immediately missed a long-range penalty chance.
Tovey's exit hit the Dragons hard, and it proved a cue for Wasps to increase the tempo of a game that had rarely threatened to leave third gear.
A combination of excellent ball-retention and repeated excursions across the gain-line took its toll on the Dragons defence, and Varndell was the beneficiary as he claimed Wasps' third try.
Another missed Walder kick - his fourth from seven attempts - meant the Dragons were still within striking range.
But although they dominated much of the second period, the Dragons missed Tovey's guile, and with Lady Luck also against them the combined product was a Wasps win, albeit after a performance Tony Hanks' team would rather forget.
Pen: Tovey 2
Tries: Jacobs, Lemi, Varndell
Pen: Walder 2
Dragons: 15 Patrick Leach, 14 Adam Hughes, 13 Tom Riley, 12 Ashley Smith, 11 Aled Brew, 10 Jason Tovey, 9 Wayne Evans, 8 Toby Faletau, 7 Gavin Thomas, 6 Lewis Evans, 5 Robert Sidoli, 4 Adam Jones, 3 Nigel Hall, 2 Lloyd Burns, 1 Phil Price.
Replacements: 16 Steve Jones, 17 Hugh Gustafson, 18 Ben Castle, 19 Scott Morgan, 20 Hugo Ellis, 21 James Leadbeater, 22 Will Harries, 23 Matthew Jones.
Wasps: 15 Mark van Gisbergen, 14 Tom Varndell, 13 Ben Jacobs, 12 Steve Kefu, 11 David Lemi, 10 Dave Walder, 9 Joseph Simpson, 8 Andy Powell, 7 Serge Betsen, 6 Joe Worsley, 5 James Cannon, 4 Simon Shaw, 3 Ben Broster, 2 Joe Ward, 1 Zak Taulafo.
Replacements: 16 Tom Lindsay, 17 Shaun Ruwers, 18 Bob Baker, 19 Marty Veale, 20 Dan Ward-Smith, 21 Nic Berry, 22 Seb Jewell, 23 Richard Haughton.
Referee: Peter Allan (Scotland)
Assistant referees: Andrew Macpherson (Scotland), Rob Dickson (Scotland)