The Dragons, who lost their opening two games in the competition, got off to a flying start with 10 points in as many minutes.
Outside half Jason Tovey was their chief architect as he landed a first minute penalty and then showed off his exquisite handling skills to send centre Ashley Smith racing in unopposed for a try at the posts.
The conversion was a formality for Tovey and the visitors were under the cosh. Dave Walder then hit the upright with his first penalty attempt before getting Wasps off the mark at the end of the first quarter with a 30 metre strike.
Then came a brilliant counter attack from the visitors which began with England and British & Irish Lions lock retrieving in his own 22 and launching a move that ended with a try for acting captain Ben Jacobs at the other end.
After the game Dragons coach Turner pointed to a forward pass from Jacobs to David Lemi, and then a knock-on by Joe Simpson later on in the move, but it was a fluent piece of play from the English Premiership outfit. Walder added the wide angled conversion and the scores were tied after 23 minutes.
Tovey, who went off with a knee problem early in the second half to become the second Dragons casualty after tight head prop Nigel Hall was carried off on a stretcher with a ruptured achilles, then kicked a penalty and dropped a goal to move the home team six points clear.
Walder's second penalty success cut the gap to 16-13, but he was then guilty of missing another on the stroke of half-time.
If the Dragons were good value for their lead at that stage, they found themselves up against it in the opening 15 minutes of the second half as Wasps poured on the power up front to pave the way for tries from wings David Lemi and Tom Varndell.
Those scores put them ahead for the first time and put them in the hunt for a potential try bonus point. But the Dragons hit back strongly and twice reached the Wasps line to forced referee Peter Allan to call on the TMO.
On both occasions, the latter in the last move of the match, the verdict went against them.