After a win and two draws from their first three matches, Belgium knew the pressure was on to produce a performance befitting a side ranked fourth in the world.
The hosts, who reached the quarter-finals of the 2014 World Cup, were 2-0 up at halftime although they could easily have been four or five ahead against a packed but often porous Cypriot defence.
Manchester United midfielder Fellaini opened Belgium's account midway through the first half after the visitors failed to clear a corner. A cross by Eden Hazard then set up Christian Benteke to make it 2-0 in the 35th minute.
Fellaini found the top corner with a stunning long-range strike in the 66th minute but that effort was surpassed a minute later by Hazard.
Debutant Michy Batshuayi completed the demolition in the 80th minute after the hardworking Radja Nainggolan dispossessed a defender.
Cyprus, who beat Bosnia 2-1 in September, were without six injured players, barely mounted an attack and a single off-target header in the second half proved their only threat.
On Tuesday, Belgium play Israel who were knocked off the top of the group after a 3-0 home defeat by Wales on Saturday.
Belgium have eight points from four games, Israel have nine from four while Wales are on 11 points having played five matches. (Reporting by Philip Blenkinsop, editing by Tony Jimenez)