Against the run of play, the Czechs earned their first point of the competition with two late strikes from substitutes -- the first time at Euro 2016 that a team has come back from two goals down.
Tomas Necid scored the equaliser from the penalty spot deep into stoppage time after Domagoj Vida was penalised for a handball.
Goals from Ivan Perisic and Ivan Rakitic had given Croatia a well-deserved 2-0 lead before Milan Skoda ensured a nervy finish when he pulled one back in the 76th minute with a powerful header from captain Tomas Rosicky's perfectly judged cross.
The game at Geoffroy Guichard stadium in Saint Etienne was stopped for a while before the Czech equalizer by Mark Clattenburg when flares were thrown onto the pitch from the Croatian fans' end and fights broke out in their stand. Ivan Rakitic and Croatia captain Darijo Srna were among those signaling to the fans to calm down.
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Speaking through a translator, Cacic described those responsible for the incidents as "sports terrorists" and said the Croatian football authorities can't fight against these "hooligans" on its own.
"They don't deserve to sit in stands," he said. "They ruin everything we are doing. It's a group of maybe six, seven or 10 individuals. I hope we can identify them and that they will be punished."
"My players lost their strengths, they might have been worried to see their relatives in the stands," Cacic said. The late reverse undid all Croatia's good work during a match it dominated with midfielders Luka Modric and Ivan Rakitic dictating the tempo with their quick passes and pinpoint crosses.
Czech goalkeeper Petr Cech was under pressure almost from the kickoff after failing to clear a back pass properly. Croatia forward Mario Mandzukic then went close with a header that Cech tipped over the bar.
But it wasn't until the 37th minute that Croatia took the lead when Perisic picked up the ball, produced a fine step-over before firing a low angled shot into the corner of the Czech net.
"It's a strange situation, we were a much better team and should have scored two, or three more goals to take the three points," Rakitic said, before condemning the violent Croatian fans.
"I think most of our supporters are true supporters. But just ten individuals can make problems," he said. "The Croatian FA is fighting against this. We just have to say sorry. Hopefully UEFA can understand this. In our next qualifiers we'll have to play in empty stadiums because of stupid supporters."