The 21-year-old took three for 31 from eight overs to help restrict the West Indies to a modest total of 169 for nine.
That target proved a little trickier than initially thought. The visitors were nicely placed at 164 for five in Antigua.
Maddy Green had anchored the chase with 48 from 78 balls on a slow surface. But her departure triggered a dramatic collapse.
Brooke Halliday (24), Lea Tahuhu (0) and Isabella Gaze (1) were all levered out in quick succession.
Otago all-rounder Hayley Jensen (5 not out) completed the win with a boundary.
Green was named player of the match, but Carson made an impressive debut.
Among her three victims was the key wicket of West Indies captain Hayley Matthews for 46. The hard-hitting all-rounder was trapped lbw after retreating deep in her crease and trying to work the ball into the leg side.
Carson had a breakthrough season for Otago last summer. She was the leading wicket taker in both the Hallyburton Johnstone Shield the Super Smash tournaments, and was rewarded with a national contract.
New Zealand lead the ODI series 2-0 after winning the opening game by five runs under the Duckworth-Lewis scoring system when bad light stopped play.
The final game of the series is on Sunday, but the two teams will play in a five-game T20 series following that.
