Wales produced a disciplined display to make it six games without defeat, and claimed the victory thanks to Rhyl-born Evans' first-half strike.
John Toshack also gave debuts to Jack Collison and Sam Vokes, while welcoming Craig Bellamy back into the fray after seven months of injury anguish.
Manchester Cuty youngster Evans struck soon after replacing unlucky skipper Carl Fletcher, who had been carried off with ankle ligament damage.
West Ham's teenage midfielder Collison made his Wales debut from the start in the Laugardal Stadium, becoming Toshack's 30th new cap in his three and a half years as national manager.
And Crystal Palace midfielder Fletcher captained the side for the first time - sadly for just 39 minutes before his injury - while Swansea's new signing Ashley Williams won only his second cap in defence.
Toshack used Chris Gunter and Joe Ledley as wing-backs, while Wolves youngster David Edwards played only his fourth game for his country as a support striker behind Freddy Eastwood.
Bellamy was on the bench - his first Wales involvement following a season of injury problems.
The Welsh started with just Eastwood up front, and Jason Koumas providing the passing quality from midfield.
The intention was to get Edwards forward in support of his Wolves team-mate, but the supply line of passes was limited, and he was soon back in a more orthodox midfield role.
There were few early chances but Iceland almost got Palmi Palmason clear on the right for a near-post effort, and after 12 minutes Wayne Hennessey produced a fine reflex save to thwart Palmason again.
Collison and Fletcher both had efforts from the edge of the box charged down as Wales sought to hit the home side on the break.
But there was a lack of tempo to Wales' game, although Koumas managed to pop up in dangerous positions, producing one cross from the right that saw Fletcher fire in a shot that was deflected wide.
But Fletcher's first game as Wales skipper was sadly to end after just 39 minutes when he was brought down from behind by Palmason.
He was carried off on a stretcher with ankle ligament damage and replaced by Evans, winning his first senior cap.
Wales reverted to a 4-4-2 system with Edwards dropping into midfield and Evans joining Eastwood up front.
Gunnar Porvaldsson could have broken the deadlock after 41 minutes with a shot on the turn that flew over the bar.
But within five minutes of coming on, Evans put Wales ahead with a neat backheel from Ledley's cross.
Fletcher's injury was confirmed during the break, and he will miss Sunday's match in Rotterdam against Holland.
Iceland sent on second-choice keeper Fjalar Porgeirsson for Sturluson at the interval.
Koumas saw a 30-yarder fizz just wide before Wales sent on Vokes, also for a his debut, after 48 minutes for Eastwood. At the same time Neil Eardley replaced Ledley.
The Under-21 partnership of Vokes and Evans was now in tandem up front, with Owain Tudur Jones coming on for Collison on the hour mark. Iceland sent on Hannes Sigurosson and Helgi Danielsson for Eggert Jonsson and Stefan Poroarson.
After 62 minutes, Bellamy came on for his 50th cap in place of Edwards, playing on the right.
Bellamy's arrival brought pace and attacking ideas, and one cross almost produced a goal for Vokes at the far post.
Iceland sent on Theodor Bjarnason for Atli Podarinsson and Jonas Saevarsson for Aron Gunnarsson with 20 minutes left as Iceland fought to get back into the game.
Palmason's stabbed effort after 79 minutes was their only chance of the second half.
Arnor Smarason replaced Porvaldasson with eight minutes left, but still Wales' defence looked calm and in control.
With a minute left, Andrew Crofts came on for Koumas, who had held the captain's arm band, and it was finally returned to Bellamy for the final moments of an impressive Wales display.