Right-hander Fletcher's top score, spiced with three fours and three sixes, led the home team to 165-6 off 20 overs at Windsor Park. Lendl Simmons contributed 36 and Trent Boult led the Kiwi bowlers with 2-22 yesterday.
New Zealand's pursuit got a spirited start through Kane Williamson (37) and Brendon McCullum (21) but the visitors were eventually bowled out for 126 off 19.1 overs.
Left-arm pacer Cottrell took 3-28 while off-spinner Narine grabbed 2-19 from four miserly overs.
Simmons and Fletcher, who notched his second straight half century, set up West Indies' competitive total after they were sent in. The right-handed pair shared a second wicket stand of 66 in 52 balls.
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There were aggressive contributions in the lower order as well as the West Indies smashed 56 runs off the last five overs.
Captain Darren Sammy hit 10 off three balls and Andre Russell ended 14 not out off seven deliveries. New Zealand lost Jimmy Nessham (7) at 8-1 when he slashed Cottrell to backward point. But Williamson, captain for the day, shared successive stands with McCullum and Taylor to boost the Black Caps.