Slovakia were making their World Cup debut as an independent nation
Slovakia coach Vladimir Weiss described conceding a late equaliser to New Zealand as a "small sporting tragedy" for the World Cup debutants.
Robert Vittek's second-half header had given Slovakia the lead but Winston Reid headed home in the 94th minute for a dramatic injury-time leveller.
"We were the better team over the game. It's a pity we didn't take the opportunities we had," said Weiss.
"The mood in the dressing room is very sad but that is football."
Slovakia, who were making their World Cup debut as an independent nation, enjoyed plenty of possession but were unable to get the better of an organised New Zealand.
Weiss's side paid for sitting back on their lead when Kiwi defender Reid rose to head Shane Smeltz's left-wing cross beyond goalkeeper Jan Mucha.
"We have to cope with the sadness and I hope we will perform well in the next match," said Weiss, whose team play Paraguay on Sunday.
The result in Rustenburg leaves Group F finely poised with all four teams on a point after Italy drew with Paraguay on Monday.
Goalscorer Vittek admitted the draw felt like a defeat. "It is so disappointing to lose a goal in the 94th minute," he said.
"It is just a nightmare. We could have had three points and at the end it felt like we have lost the match."