The hosts made it three wins from three games in Group I thanks to goals from Dries Mertens, Timothy Castagne and Romelu Lukaku.
Hazard was playing his first match since leaving Chelsea for Real Madrid and teed up the opening goal with a typical piece of sharp movement and a precise cross from the right flank.
Mertens was the recipient of Hazard's service and he turned provider as Castagne made it 2-0 soon after and Lukaku put the game beyond any doubt early in the second half.
To the delight of 36,000 at the King Baudouin Stadium, Belgium dominated throughout and only had to wait until the 11th minute for the opener, Mertens fortunate as his shot from Hazard's delivery took a telling deflection over goalkeeper Dmytro Nepohodov.
It was already clear Nepohodov was in for a busy night, after earlier efforts from Axel Witsel and Kevin De Bruyne, and the Belgium lead was doubled when Mertens found space on the left and his cross was headed home emphatically by defender Castagne, who celebrated his first international goal.
Witsel, Lukaku and Mertens all rained shots at the Kazakhstan goal before Jan Vertonghen put a difficult chance over the bar and De Bruyne rattled the crossbar as Belgium threatened a rout.
They made sure of the points in the 50th minute. Mertens cut in from the right flank and curling a left-footed shot against the base of the far post but the ball bounced out straight to Lukaku, who prodded into an empty net for the easiest goal of his Belgium career.
Belgium could have scored six or seven on another night. Hazard, De Bruyne and Mertens picked apart a weak defence relentlessly as one of the likely favourites for Euro 2020 proved their class once again.
Still keeping good Kompany
Will Vincent Kompany wind down his on-pitch career after taking on his player-manager role at Anderlecht? Belgium must hope not as he produced a performance that suggested he deserves to remain a key presence for his country, whipping a chance away from Toktar Zhangylyshbay in an excellent piece of defending.
Mertens misses his mark
Napoli forward Mertens was key in creating openings but spurned chances to add to his goal. One in particular stood out, when he skied the ball 10 yards over from barely six yards out.
Belgium host Scotland on Tuesday while Kazakhstan have a golden opportunity to claim their second win of qualification against minnows San Marino.