Wasps back-rower Andy Powell tries to break through Toby Faletau's tackle
Wasps needed a second-half fightback to overcome the Dragons in Cardiff and keep the pressure on Toulouse at the top of Heineken Cup Pool Six.
Ashley Smith's fine try and the boot of Jason Tovey gave Dragons an early 10-0 lead, before Ben Jacobs's disputed try and Dave Walder's boot got Wasps going.
Steve Kefu sent in David Lemi after the break, while the Dragons lost Nigel Hall, Phil Price and Tovey to injury.
Tom Varndell got Wasps' final try, the Dragons ending the game on the attack.
The losing bonus point was the Welsh side's first in this season's cup and their competition is all but over, but Wasps will be seeking a four-try victory in next week's reverse fixture at Adams Park to make up their two-point deficit on Toulouse in the group.
The Dragons - playing at Cardiff City Stadium as their Rodney Parade home pitch was frozen - enjoyed an outstanding start, Tovey thumping over a first-minute penalty from 40m.
The talented fly-half's take-and-give with Aled Brew then sent skipper Smith over for a superb score that Tovey himself converted.
The visitors recovered from the early onslaught, Walder easing nerves with an 18th-minute penalty after an earlier miss.
Simon Shaw was the unlikely catalyst for a Wasps attack from their own 22.
There were strong suspicions of a forward pass on the halfway line, but the Dragons' protests went unheeded as Wasps captain Jacobs crossed for a converted try to level the scores.
Tovey again took the home side clear with a penalty then a drop-goal, but the Dragons saw Hall stretchered off with a suspected ruptured Achilles tendon and fellow prop Price leave with a rib injury.
Walder then pulled the deficit back to 16-13 with his second penalty shortly before half-time.
Wasps, with full-back Mark van Gisbergen enduring a difficult afternoon under the high ball, strung little together in terms of sustained attacking play.
Yet they took the lead for the first time on 46 minutes as Kefu's break sent Lemi in for an unconverted try.
A further blow was struck to the Dragons' hopes as the outstanding Tovey went off with an injury he picked up trying to prevent the Lemi score, his replacement Matthew Jones then missing a long-range penalty shot at goal.
Varndell notched the third Wasps try after a good period of pressure, Walder's conversion again wayward.
The Dragons dominated the final quarter, had two tries ruled out by video official Iain Ramage and ended the match encamped on the Wasps line.
But they could not stop the visitors powering to their fifth successive victory and setting up a likely pool decider with Toulouse at Adams Park in January.
Dragons coach Paul Turner: "That was probably the best we have played all season.
"Their first try was a forward pass and a blatant knock on, and then the referee has gone to the TMO in the second half and we've scored through Wayne Evans half a second later.
"Wasps are a good side if you allow them to be. I felt we worked them all day, and if we had taken our opportunities we would have won the game.
"I thought we really took it to Wasps. We created so much and we more than deserved a draw. I don't like losing, and we did everything but win the game."
Wasps rugby director Tony Hanks: "The performance was not at a level we would expect.
"We defended a lot more in the second half than we would have liked, and we made some poor decisions. Our performance, in general, concerned me.
"The Dragons forced some errors, but I lost count of how many high balls we dropped. We didn't seem to have that spark about us.
"We knew the Dragons would come at us - away wins in the Heineken Cup are tough - and we will need to improve in the return fixture next week. The Dragons will not roll over."
Dragons: Leach; Hughes, Riley, Smith, A Brew; Tovey, W Evans; Price, Burns, Hall, A. Jones, Sidoli, L. Evans, G. Thomas, Faletau.
Replacements: M Jones for Tovey (47), Gustafson for Price (38), Castle for Hall (38), Morgan for A Jones (66).
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