Roach took 5-60 with a fine spell of pace bowling which ripped the heart out of New Zealand's middle order and then Gayle, who struck 150 in the first innings, eased the Caribbean side to their target of 102 with a solid, unbeaten 64.
It was just the third win for West Indies in 34 Tests since their victory over England in Jamaica in February, 2009.
New Zealand's hopes of batting out the day, or at least setting the hosts a challenging target came undone with West Indies seamers Roach and Ravi Rampaul doing the damage.
Resuming on 199 for three, after losing Brendon McCullum for 84 late on Saturday, the Kiwis lost two crucial wickets for just 27 runs before lunch.
After a short rain break, paceman Roach trapped Ross Taylor (21) lbw and then clean bowled Kane Williamson with a superb delivery that straightened and removed the right-hander's off stump.
The impressive Roach struck again shortly after lunch, with the third delivery with the new ball, luring Neil Wagner into an edge to keeper Denesh Ramdin.
Rampaul then removed Dean Brownlie and Daniel Vettori, spinner Sunil Narine claimed his eighth wicket of the match by getting Doug Bracewell lbw.
Kruger van Wyk had provided the only real resistance but stuck with number eleven Chris Martin he had to go for his shots and was undone by a lovely slower bowl from Roach.
The total of 272 all out left West Indies with a comfortable target of just 102 to win and plenty of time to get it.
Their first test win since beating Bangladesh in Dhaka in Oct 2011, ending a nine test winless streak, was never in doubt as Gayle took charge with Kieran Powell who made 30 before he was caught by Brownlie off Bracewell.
Assad Fudadin accompanied Gayle to the finish line as West Indies enjoyed the rare taste of victory and handed New Zealand another painful loss after their defeat in the one-day series.
"We didn't look like creating enough pressure at times," said skipper Ross Taylor.
"We have to be smarter in the way we play, bowl in better areas and our batsmen have to spend a lot of time at the crease," he said, looking ahead to the second Test in Kingston, Jamaica which starts on Thursday.
West Indies skipper Darren Sammy said Gayle had made the perfect return to test cricket after 19 months out during a dispute with the West Indies Cricket Board.
"With Chris back, he brings his experience and you could see that in the way he nurtured Kieran Powell along to his first Test century," said Sammy.
"We have to just keep our focus in this series and look to repeat what we did," he added.