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, who ended the innings with 32 balls to spare when he yorked last man Corey Collymore for nought, took three for 31 and left-arm spinner Daniel Vettori, who late on had two wickets in two balls, three for 39.
Impressive wicket-keeper Brendon McCullum led a fine fielding effort with four catches.
Chris Gayle (44) and captain Brian Lara (37) were the only batsmen to make headway during an innings where left-arm pace bowler James Franklin, whose three overs cost 29 runs, was alone in being expensive
Gayle and Shivnarine Chanderpaul struggled early on in the face of accurate fast bowling from Bond, who swung the ball away late at sharp pace and beat the outside edge following captain Stephen Fleming's decision to field first.
Gayle, who'd made a laboured two during West Indies' 103-run defeat by world champions Australia in their Super Eights opener, broke free by going down the pitch to off-drive Bond for four.
But at the other end Chanderpaul's agonising 28-ball stay for four ended when he edged a full-length Bond delivery and Scott Styris took a good low catch at second slip.
West Indies were now 14 for one and two runs later were lucky not to lose Gayle, then on seven, when Peter Fulton at square leg was unable to hold a diving one-handed chance off a pull against Mason.
Bond, after a fine opening spell of one wicket for 13 runs in five overs, was replaced by Franklin.
Not the fastest, Franklin can suffer if the ball is not swinging and Gayle went after him with three boundaries in as many balls, the best when no fieldsman moved after a backfoot drive sped through extra-cover.
Oram, however, maintained New Zealand's grip on the game 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 in the 23rd over 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 Scott Styris then deceived Lara with a slower ball and an edged drive was well-held by McCullum, standing up.
Dwayne Smith, bowled around his legs, and Daren Powell, plumb lbw, both fell to Vettori and although Collymore survived the hat-trick he did not last long. West Indies were playing for a third straight day after their match against Australia spanned Tuesday and Wednesday because of rain.
This was New Zealand's first Super Eights match.
The Black Caps were without batsman Lou Vincent, forced out of the World Cup on Sunday after having his wrist broken by Bond when batting in the nets.
Fulton was due to open alongside Fleming with Hamish Marshall drafted into the top order barely two days after arriving in Antigua.
West Indies 177 all out in 44.4 overs ( Chris Gayle 44, Brian Lara 37; Oram 3 for 23, Bond 3 for 31, Vettori 3 for 39) Vs New Zealand (yet to bat).