The PSV Eindhoven striker broke the deadlock eight minutes before half-time and doubled his tally just after the hour to put Slovenia firmly in control of the match between Group E's two bottom teams.
Matavz put Slovenia ahead eight minutes before half-time, latching onto Zlatko Dedic's pass and slotting his left-footed shot into the bottom-left corner of Tasos Kissas' net.
The same player grabbed his second goal in the 61st minute with a neat finish from Cesar's cross.
Aloneftis was introduced in the 79th minute and took only four minutes to make his mark, firing a left-footed shot into the bottom-left corner from 25 yards.
The APOEL winger was then fouled by Cesar, the Chievo defender seeing red for his second bookable offence, but there was not enough time remaining for Cyprus to make their extra man tell.