Valdet Rama fired the visitors ahead on nine minutes but Iceland's recovery took just four minutes to begin.
Birkir Bjarnason stabbed in a low cross at the back post four minutes later, fighting off Eidur Gudjohnsen to do so, before another close-range finish from Kolbeinn Sigthorsson just after half-time settled the match.
Victory moves Iceland into second place in Group E, five points behind leaders Switzerland but, more importantly, one point clear of nearest rivals Slovenia and two better off than Norway with two games to play.
Iceland will meet Cyprus at home next month before rounding out their campaign with a crucial away tie at Norway on October 15.
The Iceland fans had to withstand an early scare, however, when Rama netted from 20 yards shortly after Gylfi Sigurdsson had hit the post from long range.
Iceland's response was swift, and after good build-up play down the right pulled the away defence out of position, it was left to Bjarnason and Gudjohnsen to squabble over who would turn in the equaliser.
The home side continued to press and were worth their win when Birkir Mar Saevarsson's low cross from the right was steered in by Sigthorsson.