King and Brooks shared a 102-run opening partnership as the West Indies struck a winning formula at last.
After making the bold decision to rest captain Nicholas Pooran, the hosts also played more than one spinner for the first time in the series and their bowlers flourished on a slow pitch at Sabina Park.
New Zealand made only 145-7 after winning the toss, falling well short of its 185-5 in the first match and 215-5 in the second.
Glenn Phillips top scored with 41 from 26 after the Black Caps won the toss and decided to bat for the third time in the series. All of their top six batsmen got starts with multiple boundaries but neither of them pushed on to a total that could have got New Zealand past 160 and set a more compelling chase for the hosts.
The West Indies reached its target with an over to spare, as stand-in captain Rovman Powell hit Jimmy Neesham for six to clinch the win and leave the series 2-1 in New Zealand's favour.
Left-armer Akeal Hosein, playing for the first time in the series, took 2-28 while right-arm fast bowler Odean Smith took a career-best 3-29 and Dominic Drakes, also appearing for the first time, took 1-19 from four overs.
King, who missed the first two matches of the series with injury, provided impetus for the run chase by taking 53 runs from 35 balls.
Brooks carried his bat throughout the innings, finishing on 56 while Powell hurried the end by striking 27 from 15 balls.
"It was very special being able to do this at home in front of everybody," King said.
"The support is incredible, my family is here and I'm happy to be able to deliver. I think consistency is the key. When you're an opening batsman you try to give your team a good start.
"Today credit has to be given to the bowlers. They did an excellent job for us and made my job that much easier."
New Zealand failed to replicate its performances of the first two victories when it made strong starts and built partnerships around Kane Williamson who made 47 in the first match and Phillips who made 76 in the second.
Today, Hosein came on in only the third over and immediately knocked over Martin Guptill (15) to leave New Zealand 18-1. Devon Conway (21), Mitchell Santner (13) and Williamson (24) all made starts but struggled to impart momentum to the innings. That task fell again to Phillips who made 41 from 26 balls and was Man of the Series.
Phillips needed to bat through but was out at a critical stage when New Zealand was 125-5 in the 17th over. New Zealand made big finishes in the first two matches, putting on 64 in the last five overs of the first two matches but managing only 39 in the last five overs this time.
The West Indies innings stood in complete contrast with the opening partnership between King and Brooks occupying 13.1 overs and making the result a foregone conclusion.
King reached his half century from 30 balls and Brooks hung around to anchor the innings, posting his 50 from 52 deliveries.
"The West Indies adapted to the conditions nicely," Williamson said.
"With the bat we felt if we could get another 15 or so we could get a bit of momentum but they were able to take wickets throughout an execute their plans beautifully.
"They did the same thing with the bat and we couldn't open up an end with the ball. We knew that with 140 things would have to go well for us with the ball and they outplayed us today."
The teams meet in the first of three one-day internationals on Thursday.
