Despite being without Mario Mandzukic, who was only fit enough for a seat on the bench due to a hamstring strain, the hosts cruised to victory on a cold night in Zagreb, putting them within touching distance of a place at the finals in Russia next year.
Having had a hand in the first goal, forcing a foul from goalkeeper Orestis Karnezis to win a penalty that was converted by Luka Modric, Nikola Kalinic steered home a second from close range.
Greece briefly responded with a header from Sokratis Papastathopoulos, only to leak a third before the break when Ivan Perisic - left alone at the far post - headed home.
Andrej Kramaric's third goal in his last two international appearances early in the second half extended Croatia's advantage even further - leaving their opponents with a mountain to climb ahead of the return fixture on Sunday.
Greece's issues were all of their own making, with any plan to keep the tie close ahead of the return leg disappearing when they gave away two avoidable goals in the opening 20 minutes.
Karnezis' decision to take a touch with the ball at his feet backfired spectacularly when he presented possession to Kalinic, with the goalkeeper compounding the initial error by carelessly flattening the Croatian forward inside the area.
Modric converted from the spot – sending Karnezis the wrong way – and the home fans were soon celebrating again when an unmarked Kalinic converted Ivan Strinic's low cross from the left wing.
Reeling on the ropes after taking Croatia's one-two on the chin, Greece briefly countered when Sokratis' looping header from a corner cut the deficit in half.
However, having worked so hard to grab an away goal, Michael Skibbe's side conceded again within five minutes, Sime Vrsaljko's deep cross from the right presenting Perisic with the simple task of nodding the ball in.
The gap would have been greater at the break were it not for Karnezis, who went some way to redeeming his earlier error with a fine double save to deny Kramaric and Kalinic in quick succession.
Yet Greece's calamitous defending was not just confined to the first half.
Konstantinos Stafylidis' attempt to chest a deep cross back to his own goalkeeper served up a fourth goal on a plate, the alert Vrsaljko nipping in to square the ball across for the waiting Kramaric to convert.
Sokratis headed over under pressure from Dejan Lovren in a rare moment of concern for Croatia, while Danijel Subasic was required to turn away a 90th-minute header from Konstantinos Mitroglou.
Perisic wastefully blazed over when well placed at the other end as Greece narrowly avoided conceding what surely would have been the knockout blow. As it is, they still require a remarkable turnaround in the home leg in Athens.