For the West Indies this was a second comprehensive defeat in as many days after their 103-run reverse against reigning champions Australia and now the hosts face a major struggle to qualify for the semi-finals.
Chasing 178 to win, New Zealand finished on 179 for three with more than 10 overs to spare.
Styris was 80 not out - his third fifty of this tournament - with Craig McMillan unbeaten on 33 as the experienced duo shared an unbroken fourth-wicket stand of 102.
Victory had been all but assured by a disciplined bowling display that saw New Zealand dismiss the West Indies for 177 with 32 balls left in the innings.
Oram took three for 23, his treble behind a slump that saw three top-order wickets lost for 15 runs in six overs.
Fast bowler Shane Bond had three for 31 and left-arm spinner Daniel Vettori, who late on took two wickets in two balls, finished with three for 39.
Impressive wicket-keeper Brendon McCullum led a fine fielding effort with four catches.
In reply, New Zealand's Peter Fulton, promoted to open after Lou Vincent's World Cup ended when his wrist was broken batting against Bond in the nets on Sunday, was bowled second ball by Daren Powell.
Hamish Marshall, Vincent's replacement, then fell for 15 when he chipped Powell to West Indies captain Brian Lara at mid-off.
But left-hander Fleming, driving confidently and lofting Corey Collymore for six over square leg, received solid support from Styris.
Fleming was though run out by yards for a 66-ball 45 after Lara's direct hit from mid-wicket on the striker's end stumps.
New Zealand were now 77 for three but McMillan stopped any West Indian momentum that may have been building with three aggressive boundaries off seamer Dwayne Smith.
Styris subssequently completed a 72-ball fifty with three fours.
Earlier Chris Gayle (44) and Lara (37) were the only West Indies batsmen to make headway following Fleming's decision to field first.
The new ball duo of Michael Mason and Bond ensured runs were hard to come by early on with some accurate swing bowling and it was only the introduction of left-armer James Franklin, whose three-overs cost 29 runs, that released the pressure.
Oram, however, maintained New Zealand's grip when Ramnaresh Sarwan (19), was superbly caught left-handed by a diving McCullum after the batsman inside-edged an intended cover-drive.
The giant all-rounder struck again when Marlon Samuels (nine) was caught behind off the glove before he had Gayle, chopping onto his stumps for 44 made in 56 balls with eight boundaries.
West Indies were now 81 for four with, as happened against Australia, none of the top four making a big score.
Lara, nearly run out on eight by McCullum's throw to the bowler's end, and Dwayne Bravo rebuilt steadily while adding 47 for the fourth wicket.
But their partnership was broken when Bond's first ball back saw Bravo caught behind for 18.
Canny medium-pacer Styris then deceived Lara with a slower ball and an edged drive was well-held by McCullum before the tail came and went.
West Indies, playing for a third straight day after their match against Australia spanned Tuesday and Wednesday because of rain, face Sri Lanka in Guyana on Sunday and 24 hours later New Zealand take on Bangladesh in Antigua.