McCullum's 51 was the standout effort in the New Zealand innings, while too many others didn't get starts. It took a 44-run stand for the final wicket between Nathan McCullum and Mitchell McClenaghan to give New Zealand something to bowl at.
Later, McClenaghan's five for 58 ripped the top off the West Indian innings and are his best ODI figures.
"The win was very important at the moment," West Indies captain Dwayne Bravo said.
"The confidence had been low as a team and that's going to take a lot of pressure off us, and me as captain.
Bravo was delighted with the bowling and fielding performance but acknowledged the West Indies batting needed plenty of work.
Even at 96 for six he was optimistic, given "one of the cleanest hitters in world cricket" - Sammy - was at the crease.
"I'll be honest and say there were some nerves settling in but I know as long as Sammy is there we still had a good chance."
The second ODI is in Napier on Sunday.
SCOREBOARD