First-half goals from Borek Dockal and David Lafata gave Pavel Vrba’s side a commanding advantage which was added to by Lukas Vacha.
Kazakh defender Yuri Logvinenko scored two consolation goals in the final 10 minutes but Tomas Necid’s strike meant the visitors were never concerned.
They took the lead in the 13th minute when Dmitri Miroshnichenko’s dismal clearance fell to Daniel Kolar, who set up Dockal for a low right-foot finish from 20 yards.
Sergei Khizhnichenko’s turn and shot at the other end quickly provided a first test for Petr Cech but lacked the power to beat the Chelsea man.
The Czechs were denied a delightful second goal when after a fluent passing move, Pavel Kaderabek crossed and Ladislav Krejci headed in at the far post but was narrowly offside.
Khizhnichenko dragged a left-foot shot well wide before his deft backheel almost set up Tagybergen, who was just unable to stretch and meet the ball.
But Kazakhstan gave up another shambolic goal just before the interval when David Limbersky’s cross saw four Czech players running in unopposed on keeper Andrey Sidelnikov. Kaderabek stretched to hook the ball across and give Lafata an easy finish.
Cech’s clearance was almost charged down by Khizhnichenko and Tagybergen lashed an over-excited effort wide before Logvinenko stretched to poke a shot onto the outside of the post.
But any attacking momentum for Kazakhstan was once more undone by defensive failings. Again Limbersky’s cross was allowed to travel too far and again the man at the back post was unmarked, Vacha volleying low past Sidelnikov.
Nusserbayev got a slight touch to Tagybergen’s low shot but could not test Cech and substitute Ulan Konysbaev curled just wide.
Khizhnichenko missed a good headed chance and Miroshnichenko’s dangerous cross eluded Tagybergen.
An isolated Nusserbayev lashed wastefully into the side netting and Tagybergen’s leaping volley flew over before the hosts finally got on the scoresheet.
From Bauyrzhan Islamkhan’s left-wing corner, Dockal inadvertently headed the ball straight to Logvinenko who finished from six yards.
But Necid restored the Czech’s three-goal margin with a fine turn and finish from the impressive Limbersky’s pull-back and a towering header from Logvinenko merely added further Kazakh consolation.