Earlier, Sammy had won the toss and chose to bat at Warner Park and although Southee managed to dismiss dangerman Chris Gayle cheaply for the second game in a row, West Indies were able to post a solid total.
Kieron Pollard (56) and Marlon Samuels (46) top-scored for the Calypso Kings, while Jacob Oram picked up 3-42 from 10 overs to be the pick of the bowlers for the visitors.
Southee also ended up with three scalps as he took the last wicket as the West Indies were bowled out in the final over but the Kiwis didn't help themselves as they bowled 16 wides.
New Zealand's chase got underway slowly when they lost Martin Guptill in the first over for a duck before Rob Nicol went about steadying the ship as he smashed 35 from 32 balls.
Kane Williamson and Brendon McCullum went without providing too much work for the scorers and New Zealand looked on the ropes at 75-4 before Tom Latham and Taylor built a strong partnership.
They produced 71 for the fifth wicket before Latham went for 32 and Taylor steadily ran out of partners before he was eventually removed himself.
The fifth one-dayer will be played at the same venue on Tuesday. (NZT)