Oram took three top-order wickets in the space of six overs with his deceptive bounce and late movement off the seam. The tournament hosts did not recover and eventually succumbed for 177 from 44.4 overs.
Captain Brian Lara did his best to shore up his team's fragile batting for the second time in two days after the opening second-round game against Australia spilled over into the reserve day.
But when he was caught behind by Brendon McCullum from a Scott Styris off-cutter for 37 any hope of West Indies posting a competitive total disappeared.
After Stephen Fleming had asked West Indies to bat following a morning shower, Shane Bond made the new ball swing extravagantly away from left-handed openers Chris Gayle and Shivnarine Chanderpaul. He was soon rewarded when the latter was caught by Styris at second slip for four.
Gayle on seven smashed a ball through Peter Fulton's fingertips diving to his right at square-leg and then took three fours in a row off James Franklin's straight-forward left-arm medium pace.
The introduction of Oram, who said he would have the ring finger he broke taking a catch against Australia this year cut off if necessary to play in the tournament, was the turning point.
Oram had Ramnaresh Sarwan (19) brilliantly caught by the diving McCullum who took an inside edge in his left glove.
Marlon Samuels (9) gloved a catch to the wicketkeeper and the dangerous Gayle, who had scored 44 from 56 balls with eight fours, dragged a ball on to his stumps.
Lara's dismissal to Styris's clever medium-pace spelled the end of West Indies' prospects and two wickets in as many balls by left-arm spinner Daniel Vettori hastened the end.
Scoreboard
WEST INDIES:
C.H. Gayle b Oram 44
S. Chanderpaul c Styris b Bond 4
R.R. Sarwan c McCullum b Oram 19
M.N. Samuels c McCullum b Oram 9
B.C. Lara c McCullum b Styris 37
D.J. Bravo c McCullum b Bond 18
D. Ramdin c Oram b Vettori 15
L.M.P. Simmons not out 14
D.R. Smith b Vettori 8
D.B. Powell lbw b Vettori 0
C.D. Collymore b Bond 0
EXTRAS (B-1, LB-5, W-3) 9
TOTAL (all out 44.4 overs) 177
FALL OF WKTS: 1-14 (Chanderpaul, 7.5 ov), 2-66 (Sarwan, 16.2 ov), 3-78 (Samuels, 20.3 ov), 4-81 (Gayle, 22.2 ov), 5-128 (Bravo, 33.1 ov), 6-150 (Lara, 36.3 ov), 7-158 (Ramdin, 41.3 ov), 8-176 (Smith, 43.3 ov), 9-176 (Powell, 43.4 ov).
BOWLING: Mason 6-2-14-0 (1w); Bond 8.4-0-31-3 (2w); Franklin 3-0-29-0; Oram 8-2-23-3; Styris 10-1-35-1; Vettori 9-1-39-3.
NEW ZEALAND: S.P. Fleming, P.G. Fulton, H.J.H. Marshall, S.B. Styris, C.D. McMillan, J.D.P. Oram, B.B. McCullum, D.L. Vettori, J.E.C. Franklin, S.E. Bond, M.J. Mason.
UMPIRES: R.E. Koertzen (South Africa) and Asad Rauf (Pakistan).
TV UMPIRE: Aleem Dar (Pakistan).
MATCH REFEREE: M.J. Procter (South Africa).—Agencies