The Repre are the only country in the tournament making their debut, while the All Whites are making their second appearance in the competition having lost all three of their matches in 1982.
Slovakia were expected to win the match comfortably but looked nervous in the opening stages, with the Kiwis creating the first few chances with dangerous long balls pumped into the box by Tony Lochhead.
It wasn't until the 22nd minute that Slovakia threatened when Marek Hamsik had a shot that curled wide of the post from just inside the penalty area, after clever play from Manchester City's Vladimir Weiss.
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The majority of the Repre's attacks came through Weiss and after 29 minutes he created another opportunity with a couple of one-twos before slipping a pass through to Stanislav Sestak whose effort failed to trouble Mark Paston in the All Whites goal.
Paston looked shaky and he nearly gifted Slovakia a goal with a mis-kick on 34 minutes that found its way to Robert Vittek but the striker was unable to capitalise from the goalkeeping error.
The second half got off to a lively start, as Slovakia attacked from the outset and, on 50 minutes, they got the breakthrough. Sestak got down the right and whipped in a ball for Vittek to glance a header into the far corner after a lapse of concentration from Reid in the All Whites defence.
Despite the goal New Zealand had the majority of possession but failed to create any goalscoring chances, while Slovakia looked dangerous playing a counter-attacking game.
On 69 minutes a good break from the Slovaks led to Sestak getting into the box but Reid was able to get across in the nick of time and get a vital block as he put the ball out for a corner.
It wasn't until the 88th minute that the All Whites caused a threat on goal and they played to their strengths with a fantastic long ball whipped into the box by Lochhead, but Shane Smeltz could only head wide with the goal gaping.
However, the same tactic paid off in stoppage time when they put pressure on the Slovakian goal. Another long ball into the box was met by the head of Reid, and the central defender made no mistake from eight yards to give the All Whites their first ever World Cup point.