New Zealand dominated the first three-and-a-half days of the Test before a Darren Bravo double century inspired a West Indies fightback at the University Oval.
The hosts slumped to 4-44 but Ross Taylor and Corey Anderson put them back on track before the rain forced players off just before tea.
Shane Shillingford ended with 4-26 after claiming just one wicket in the hosts' first innings of 9(dec)-609.
Off-spinner Shillingford grabbed all four wickets to fall after the tourists were finally dismissed for 507 - their fourth-highest second-innings total.
Starting the day on 6-443 and a lead of 47, the West Indies, dismissed for 213 first time around, lost Bravo early as the New Zealand bowlers wrapped up the innings in the first session.
Bravo's 218, the second-highest score by a West Indian in New Zealand, and his occupation of the crease for 572 minutes took the game into the final day.
New Zealand have themselves to blame after shelling four catches in the Windies' second innings, including Bravo on 82.
The 30 overs they faced was more than enough to reach their total before the rain but a combination of good West Indies bowling, particularly by Shillingford, and lacklustre batting cost New Zealand victory.
Taylor, unbeaten on 16 at the close, was not dismissed in the Test, scoring 233 runs and edged out Bravo for the man of the match award.
Bravo's match-saving innings against New Zealand has helped continue a remarkable run of sides avoiding defeat when following on if a player scores a double century.
Bravo's 218 was compiled over more than nine-and-a-half hours and, when rain descended on Dunedin's University Oval on Saturday, he joined the ranks of six others who had passed the 200-mark when asked to bat for a second time and saved the match.
The 24-year-old left-hander, a cousin of Brian Lara, produced the kind of innings reminiscent of the West Indian great.
"Credit must go to Darren Bravo. He did exactly what was needed," captain Darren Sammy said after his side saved the Test despite a 396-run first-innings deficit.
"I guess even the great Lara would have been proud of that innings. It was a match-saving innings and we are all happy for him."
Sammy said after being written off by a New Zealand commentator as the worst West Indies team to tour the country, it was a morale boost for the side to save the Test from the position they found themselves in after the first innings.
"From the situation of being asked to follow on with a deficit of 400, it was important that we occupy the crease for long periods and I bet, on the third day, nobody expected us to be here," said Sammy, who contributed 80 as the West Indies made 507 in their second innings.
Opposing captain Brendon McCullum praised Bravo's knock but also reserved mention for Sammy's role in their fightback.
"We knew they're a team, when they're under pressure, (who) find a way to get themselves out of it. That's credit to them and credit to their leadership under Darren Sammy.
"He's done a brilliant job and showed that again in this Test match. In a time that was pretty difficult, he stepped up and did the job well," he said.
With a limited preparation for the Test, Sammy said his team failed to perform over the first two days as New Zealand racked up 9(dec)-609 after being asked to bat and being dismissed for 213 in reply.
But as the Test wore on, his players - only two of whom have toured New Zealand before - came to grips with the conditions and Sammy believes they can look ahead to the second Test starting in Wellington on Wednesday with confidence.
A year to the day since taking over as New Zealand cricket captain, McCullum is still seeking his first tick in the win column.
New Zealand were left 33 runs shy of breaking the duck for the 32-year-old after the weather, Bravo and some spilled catches allowed the tourists to get off the hook after being forced to follow on 396 runs behind.
It means McCullum has yet to taste victory in his 10 Tests at the helm and the Black Caps have twice come up short on the final day when pushing for victory.
"I thought we were excellent through the majority of that Test match and played some really good cricket especially after losing the toss on a pretty green wicket.
"To get 600 and set the game up like we did gave ourselves a real opportunity and it's disappointing it started raining," he said.
His first-innings century came after criticism of his recent Test performances and after he revealed that an ongoing back injury had forced him to consider his future in the game.
"It was relief. I don't think I've played under that much media or public scrutiny throughout my career."
The lack of Test success was starting to weigh on his mind but he felt they were denied victory by the weather.
"You're obviously judged by your results. It frustrates you, but we did everything we could to win this Test match and I believe we'd have won if it hadn't rained.
"I firmly believe we are performing pretty well in Test cricket and, sooner or later, the wins will come."