The tournament debutants had led from the 50th minute thanks to a Robert Vittek header but a failure to convert two other clear opportunities came back to bite them when the All Whites scored in the last minute of added time on Tuesday.
"It was the first match, and that is why what happened happened," defender Marek Cech told reporters.
"If it had been the second or third match, we would have coped with it."
With all the teams in Group F level on one point after Italy and Paraguay also drew 1-1, the advantage is back with the world champions and South Americans.
"I think this disappointment is justified and of course it will make things complicated," said goalscorer Vittek.
"It was a blow for us of course this goal but we will have to keep going and try against Paraguay (on Sunday) but it will be very difficult.
"We will surely miss these two points."
Coach Vladimir Weiss, whose player-son of the same name said he would struggle to sleep because of the disappointment, said they would try not to dwell on it and learn from their mistakes, which included some poor marking and hesitancy in front of goal.
"Several times we trained for special situations - these mistakes shouldn't have happened," said Weiss.
"We will play for everything, we will try to win and let's see what happens in the group.
"It's a sad mood at the moment but it also belongs to football, we have to cope with these negative emotions but I hope we will perform well in the next match."
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