Ivica Olic put the visitors ahead inside two minutes with Mario Mandzukic doubling the lead in the 32nd.
Vedran Corluka then stretched the advantage to 3-0 shortly after halftime, leaving Turkey with a mountain to climb in the return leg at Croatia on Tuesday.
Corluka set up the opener with a pass through the face of goal that took a slight deflection off the goalkeeper and fell to Olic, who rolled it home from six yards.
Turkey controlled the majority of the possession over the course of the match, but Croatia made the most of its opportunities and took a 2-0 lead through Mandzukic, who nodded a cross from Darijo Srna past goalkeeper Volkan Demirel.
Srna was involved again in the 51st minute when his free kick was played to the front of goal for Corluka, who was left wide open a few yards from the net and easily headed home the cross.
A first-half goal from Keith Andrews put the Irish on top, and Estonia was reduced to 10 men before the break when Andrei Stepanov was sent off.
Goals from Jonathan Walters and Robbie Keane three minutes apart made it 3-0, and things got worse for the hosts with the sending off of Raio Piiroja and a late penalty kick from Keane.
The Czech Republic came on strong in the final 30 minutes of its 2-0 win over Montenegro as Vaclav Pilar broke the scoreless deadlock in the 63rd minute, while Tomas Sivok tallied a second goal for his team in stoppage time.
The other match on Friday saw Portugal claim a 0-0 draw at Bosnia-Herzegovina, leaving things wide open for the return leg next week.