Resuming on 135-4 at Sabina Park, the hosts comfortably reached their target 40 minutes before lunch, finishing on 206-5.
Roach led the way with his no-nonsense approach, spanking four boundaries in a career-best innings that spanned 60 balls and 80 minutes.
West Indies won the two-match series 2-0 to follow successes against New Zealand in Twenty20 matches (2-0) and one-day internationals (4-1).
West Indies captain Darren Sammy said the challenge was to build on the success.
“It's all about being consistent,” Sammy said. “We've got the feeling of what it is to win and I know everybody in the dressing room likes that feeling so it's something we want to keep ... The last year or two has been really tough for us and we're going to enjoy this, definitely celebrate it, but that doesn't mean we're where we want to be.”
New Zealand's only success in the morning session came through off-spinner Kane Williamson, who removed Roach with just 23 needed. Roach slashed to backward point where Tim Southee pouched an easy catch.
Chanderpaul, who moved his career test average to 50.20, and Roach shared a 70-run fifth wicket partnership after coming together on Saturday at a nervous time for the West Indies on 113-4.
After Roach departed on 183-5, Guyanese left-handers Narsingh Deonarine and Chanderpaul led West Indies to victory. Deonarine lashed two fours in an unbeaten 15 before Chanderpaul struck the winning single with a glance off the back foot against Williamson.
New Zealand skipper Ross Taylor blamed inconsistent batting for his team's failures throughout a frustrating tour.
“We cannot be satisfied with 50s and 60s,” Taylor said. “We're a young team. We're getting better every now and then but we're not quite there at the moment.”
Marlon Samuels was man of the match for a dazzling 123 in the first innings and a valuable 52 in the second.
Roach, who took 12 wickets in the two tests at an average of 18.25, was man of the series.