Smith took 5-46 from his 10 overs, easily his best figures at international level, to give the tourists a real chance of avoiding a clean sweep after losing the first four matches of the series.
They would have dismissed New Zealand for even less had it not been for Styris (90), who resurrected the innings with a patient display that ended in the penultimate over when he threw his wicket away chasing a century.
New Zealand crashed to 96-4 in the 23rd over despite a 32 from skipper Stephen Fleming and a spectacular 31, which featured seven fours, from Ross Taylor, who was playing in just his second match.
Styris had a let-off on 30 when he was dropped by Chris Gayle before going on to post his 14th half-century. He eventually faced 119 balls and hit four boundaries.
Smith had only taken 15 wickets in his 34 previous ODIs but dismissed Fleming, Hamish Marshall (19), Brendon McCullum (10), James Franklin (7) and then Styris to complete his first five-wicket haul.