Set a colossal target of 531, 113 more than the current record for a successful fourth-innings chase, the West Indies were 457-6 at the end, the second-highest Test fourth innings.
Greaves battled for almost 10 hours and faced 388 deliveries, bringing up his maiden double century in the penultimate over.
Roach was unbeaten on his Test best of 58 which included facing 72 dot balls when he was on 53.
The pair put on a gutsy 180 for the seventh wicket.
“Kemar, the senior pro, guided me all the way,” said man-of-the-match Greaves.
“Coach told me once you get in, stay in. To be a part of history is exciting, but for me, it’s one day at a time.
“For us, it was just about getting into the last session.”
Given the magnitude of the chase, the West Indies saw the draw as a win while New Zealand felt they had lost.
“I’d say it feels like a win. The guys are proud,” West Indies captain Roston Chase said, adding they felt they the game was theirs going into the final session.
“But it didn’t quite work out that way that they [Greaves and Roach] wanted so when it came to the last hour we had a discussion and they said they would just play it out.”
However, the mood was not buoyant in the New Zealand camp.
“When you have a draw in that fashion where you know you’re so close to winning it does have that [losing] feeling,” captain Tom Latham said.
The second Test starts in Wellington on Wednesday. (AFP)