The co-hosts were in trouble at 22-4 when Rutherford came to the crease and continued to struggle for much of the innings, slumping to 30-5 and 76-7.
But Rutherford kept his cool and timed his onslaught to perfection as he took 37 from the last two overs of the innings to lift West Indies to 149-9 in Trinidad.
That proved to be more than enough on a two-paced surface as New Zealand fell away in the middle overs of the chase.
Fast bowler Alzarri Joseph took 4-19 and left-arm spinner Gudakesh Motie 3-25 as the Black Caps, after some late hitting from Glenn Phillips and Mitchell Santner, ended on 136-9.
While West Indies can look forward to the next stage of a home tournament, back-to-back defeats leave New Zealand’s qualification hopes hanging by a thread.
They must now beat Uganda and Papua New Guinea in their remaining games and hope other results go their way.
They could be eliminated before they play again if Afghanistan beat Papua New Guinea on Friday (01:30 BST).
Prior to the match the atmosphere at the Brian Lara Academy was one of excitement and anticipation after two victories to start the tournament for West Indies.
Six overs later, it had all fallen rather flat as the order was dismantled by New Zealand’s high class and highly experienced new-ball attack, led by Trent Boult.
In the next 12 overs the locals were stirred occasionally but, whenever West Indies appeared to be gaining momentum, a wicket fell.
Rutherford had scored only 20 after facing 21 balls but, having seen his team-mates come in, hit a boundary or two and swiftly get out, he knew he had to bide his time.
It was only when the ninth wicket went down that he really let loose, justifying his decision to turn down a single from the first ball of the 19th over by hitting three of the next four for six.
