Reid, 21, who has lived in Denmark for 11 years where he plays for top flight Midtjylland, scored his first goal on his fourth appearance for New Zealand with a well-placed far-post header with 30 seconds remaining in the Group F game.
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Pushed forward in search of an equaliser after Robert Vittek had headed Slovakia in front after 50 minutes, Reid timed his run and jump perfectly to meet an angled Shane Smeltz cross from the left with a header that went in off the foot of the post.
The goal brought joy to New Zealand fans everywhere as they celebrated their team's achievement after the All Whites had suffered three straight defeats in their only previous World Cup appearance in Spain in 1982.
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"I think the nation will be in reasonably good spirits tonight," said modest New Zealand coach Ricki Herbert. "We keep on dreaming. We have a chance like everyone else."
Asked if it was their best ever achievement he added:
"It would have to be. We have never picked up a point at a World Cup. We have thrown some really good punches tonight and got what I thought was an extremely good result".
TOTAL COMMITMENT
New Zealand were often outplayed by technically superior opponents in the second half of a poor game played in a high wind and cold conditions, but nobody could question their organisation or total commitment.
Reid, who has a Danish step-father, was in the Denmark under-21 team and national development system and had turned down a chance to play for New Zealand in 2007 before finally accepting Herbert's approach the second time.
Until Reid's late strike Slovakia, making their World Cup debut as an independent nation, had seemed set to top the group ahead of world champions Italy and Paraguay who drew 1-1 in the group opener on Monday.
Vittek struck when he met a diagonal Stanislav Sestak cross with a powerful header from 10 metres after Reid missed the ball completely in the swirling air at the Royal Bafokeng stadium.
It was a rare moment of quality in a dreary game during which New Zealand, using a 3-4-3 formation, largely frustrated the cautious Slovaks who appeared reinvigorated after advice at the interval about how to approach the second half.
The Slovaks lifted their tempo and pushed forward more aggressively from then on and gave New Zealand few chances to regain a foothold in the game, their superior technique and movement giving them control of possession.
FEW CHANCES
Slovakia had enjoyed most possession in an opening half littered with errors with few ch
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