The world champions, who have now lost two and drawn two, are bottom of Group 1 on two points, five adrift of Croatia who prevailed thanks to an early Luka Modric penalty.
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Austria are third on four points.
France coach Didier Deschamps reshuffled his team again as he looked to rest players at the end of the season and in all four matches this month never fielded the best possible lineup.
Les Bleus looked short of ideas in a defeat to Denmark plus draws against Croatia and Austria before Monday's loss to the team they beat in the World Cup final four years ago.
Mbappe had another couple of chances but Ivica Ivusic saved comfortably as Croatia controlled the pace of the game.
But it proved in vain as France failed to score for the first time after netting in their 23 previous games.
"It was a difficult month of June in terms of results. We did not have the energy and the strength against teams who had more than us," said Deschamps.
"We lost a bit of self-confidence along the way but it's mostly freshness that we lacked. When you play so many games it is very very hard."