Everything seemed to be going according to plan for the hosts when midfielder Armando Sadiku headed them in front with 10 minutes of the first half remaining.
But Cyprus were level in first-half stoppage time thanks to a 35-yard free-kick from Vincent Laban.
The second half proved to be an even affair with the hosts gaining the upper-hand towards the end of the match.
Their superiority paid off when substitute Edgar Cani headed home Jahmor Hyka's cross with six minutes remaining.
The result was secured two minutes later when fellow substitute Erjon Bogdani fired home from 30 yards.