The All Whites, who lost all three matches in their only other finals appearance in 1982, were heading for defeat after Robert Vittek had headed Slovakia ahead in the 50th minute.
But three minutes into injury time, Reid towered above a host of defenders at the back post to head Shane Smeltz's cross and deny Slovakia victory in their first World Cup match as an independent nation.
The 21-year-old Reid, 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.
Pushed forward in search of an equaliser, he timed his run and jump perfectly to meet an angled Smeltz cross from the left with a header that went in off the foot of the post.
That was the most important goal of my life," Reid said after the match. "I try to get forward but don't often score, but this is great.
"I actually didn't see the ball. I saw it late. When I saw it coming, I knew I just had to guide it on goal and not hit it too hard and it would go in. There was so much pace on the ball, I just had to guide it in."
If Reid etched his name in New Zealand football history it is only because of a Fifa rule change around eligibility allowed him to don the All White shirt.
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 New Zealand coach Ricki Herbert's approach the second time.
Until Reid's late strike Slovakia had seemed set to top the group ahead of World champions Italy and Paraguay who drew 1-1 in the group opener on Monday.
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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