Ivan Rakitic was Croatia's hero as Slaven Bilic's team secured the win they required to set themselves up for Wednesday's match with England at Wembley.
Rakitic scored midway through the first half as Croatia moved to four points adrift of leaders England in Group 6, having played one more match than Fabio Capello's team.
If England beat Croatia next week then they cannot be caught and will qualify automatically for next summer's FIFA World Cup™ finals in South Africa, but Bilic's team, for now, still have distant hopes of claiming top spot.
Perhaps more importantly in the long run, today's victory for Croatia re-opens the three-point gap between themselves and Ukraine, which for a few hours had been wiped out by Ukraine's 5-0 win against Andorra. Croatia have also played one more match than Ukraine.