The Kukis made the most of Waless errors in the first half, and led 16-4 at the break with tries from Daniel Fepuleai, Keith Lulia and Isaac John, with the solitary Welsh try coming from Rob Massam.
Despite conceding the first try of the second half from Dominique Peyroux, Wales hit back with three back-to-back tries from Lloyd White, Christaan Roets and Rhodri Lloyd to leave them two behind with 10 minutes to play, before Jonathon Ford sealed the win with Lloyd scoring a consolation try late on.
Wales struggled to get theirs hands on the ball early on and, after numerous mistakes, faced repeated sets of six and were punished when hooker Fepuleai dived over the line from dummy-half.
The home side responded well though, and after forcing Pittman into touch, Massam received Rhys Evanss pass and went over in the corner to reduce the deficit to two.
Wales looked to have carved out another opening when Peter Lupton offloaded to Ross Diverty who looked to run to the line, before play was pulled back for a forward pass from the scrum-half.
Keith Lulia, playing his last match on British soil, extended the Cook Islands' lead following a fine move when Zeb Taia found the centre from a quick play-the-ball with the Welsh too slow to react.
John, taking the ball at first receiver jinked through the Welsh defence to score the Kukiss third try on the stroke of half time.
Wales conceded another penalty as Keith Lulia broke towards the line before having the ball stripped in the tackle, with the resulting set of six bringing a try for Peyroux and a third successful conversion from Chris Taripo.
Wales gave themselves a lifeline when White dummied from the play-the-ball before diving over the line to score, before adding the extras himself to reduce the gap to 12.
The Dragons were gaining in momentum following the try, and after turning up the pressure on the Kukiss line, Lupton sought out Roets who went over to bring Wales within eight. Danny Jones hit the post from the conversion attempt to leave to remain two scores behind with 20 minutes remaining.