Substitute Wes Burns, one of seven players to win their first caps, grabbed the opener with a dozen minutes to play before Jonathan Williams and Robert Ogleby sealed a first win for manager Geraint Williams.
Wales, who will face England in their qualification group, should have led after just 15 seconds. Williams took advantage of a slip by Orri Omarsson in the visiting defence, but his curling shot clipped the top of the bar with the keeper beaten.
It proved to be a bright opening quarter from the hosts, and neat play between captain Lee Lucas and left-back Kieron Freeman saw Billy Bodin flash a volley across the six-yard box.
Some brave defending blocked efforts from Lucas and Manchester City's Emyr Huws, but as the half went on Wales became increasingly guilty of trying to be too intricate in attack.
They were nearly punished when Iceland created a first clear opening on 40 minutes as Kristjan Emilsson burst into the area only to shoot into the side-netting.
The visitors were almost gifted a goal early in the second half as Elliot Hewitt's slack backpass was seized on by Armor Traustason, but Wales keeper Danny Ward made the save.
Ward then denied Emil Atlason after the striker was played in by an excellent through ball.
But Wales regained their composure and Burns lofted the opener over the keeper after racing on to Lloyd Isgrove's pass with 12 minutes left.
Williams struck two minutes later as his smart run into the box ended with the Crystal Palace man shooting high into the net, before Ogleby tapped in the third.