American Olympic gold medallist Jamie Anderson scored 85.78 to claim victory, with Japan's Miyabi Onitsuka second.
Knight, 20, qualified second in his heat on Sunday and laid down 84.58 on his second run to finish just one point behind gold medal winner Marcus Kleveland of Norway in the final.
Under pressure after a mistake in his first run, he landed consecutive 1260s for the first time in competition to claim a first World Cup medal.
Canada's Darcy Sharpe claimed silver with a score of 85.33, with more than three points separating the top-three and American Chris Corning in fourth.
The medals are New Zealand's second and third of the Winter Games after Carl Murphy was third in the World Para Snowboard Cup LL2 category on Saturday.
- AAP