Callum Samson removed Kupakwashe Muradzi and Kian Blignaut, while Dhruv Patel fell to the bowling of Hunter Shore, leaving Zimbabwe at 54/3 at the end of the powerplay.
Two brief partnerships followed in an attempt to steady the innings, but New Zealand continued to keep a tight grip on proceedings. Harry Burns trapped Takudzwa Makoni lbw, before Luke Harrison produced a sharp caught-and-bowled dismissal to remove Simbarashe Mudzengerere. With those blows, Zimbabwe were reduced to 105 with half their side back in the pavilion.
Michael Blignaut (23) and Leeroy Chiwaula (31), helped Zimbabwe breach the 200-run mark, but they ultimately fell short and were bowled out for 218.
Harry Burns was the standout performer with the ball for New Zealand, returning figures of 3/43, while Callum Samson impressed with 2/15. Selwin Sanjay, Hunter Shore, Luke Harrison, Snehith Reddy and Jacob Cotter also chipped in with a wicket each to wrap up the innings.