Ivica Olic gave Slaven Bilic's side the perfect start when he turned home Vedran Corluka's cross from close range with just two minutes on the clock.
Mario Mandzukic doubled the lead half an hour later when he nodded past the flailing Volkan Demirel at the far post, before Corluka completed the scoring in the 51st minute with only his second international goal.
A late header from Tomas Sivok saw the Czech Republic put one foot in the finals with a 2-0 victory over Montenegro.
The Czechs held a narrow lead after Vaclav Pilar's 63rd-minute goal but Sivok's goal in injury time gives them a clear advantage going into the second leg in Montenegro.
Captain Robbie Keane claimed a double as the Republic of Ireland took a stranglehold on their play-off with a 4-0 win against nine-man Estonia.
Keith Andrews had got the visitors off to the perfect start with a 13th-minute header and Jonathan Walters repeated the feat from Keane's 67th-minute cross.
Keane made it 3-0 four minutes later and then converted a late penalty as Estonia, who had Andrei Stepanov sent off before the break and captain Raio Piiroja after it, capitulated on a miserable night for the hosts in Tallinn.
Bosnia-Herzegovina failed to make the most of home advantage as they played out a dire goalless draw with Portugal in Zenica.
Possession was dominated by the visitors but Helder Postiga missed the best of what chances they created, while Cristiano Ronaldo again struggled to recreate his club form in a Portugal shirt.
Despite that, Portugal will feel they are best placed to qualify ahead of the second leg in Lisbon on Tuesday night.