Substitute Marouane Fellaini swept home from close range in injury time to complete the scoring.
Egypt, held goalless for the second time in three matches, showed how much it misses the injured Mohamed Salah.
Driven forward by the speed and insight of Hazard and Kevin De Bruyne, Belgium was up 2-0 at halftime before losing some fluency amid a flurry of substitutions after the break.
Salah, Egypt's great scoring hope for the World Cup, is still recovering from a shoulder injury he suffered in the Champions League final as Liverpool lost 3-1 to Real Madrid. The Premier League's top scorer has been named in Egypt's squad for the tournament in Russia and coach Hector Cuper badly needs him.