Kate Cross was England's standout bowler at Basin Reserve in her first international white-ball match since September.
The seamer should have taken a wicket with the second ball she bowled when Bates chipped to mid-on, but Sophie Ecclestone dived over.
When Bates was dropped again by Lauren Bell at the start of the next over and survived after England opted not to take an lbw review soon after, the visitors could have been frustrated.
But Cross continued to bat calmly, shaping the ball well and making the most of the green-tinged surface to nibble the ball off the seam.
Despite her prolific batting, she had to wait until her second spell for her reward. New Zealand's openers struggled against the new ball.
Bates and Bezuidenhout had been tied down by England's quicks and although the introduction of spin made the runs easier, it also proved to be their undoing as Dean made the breakthrough and Ecclestone quickly took New Zealand's second wicket.
New Zealand rallied with Amelia Kerr, who captained the side for the first time in ODIs, and Georgia Plimmer.
But Cross returned to remove both lbw in an extraordinary spell that sparked a New Zealand collapse of eight wickets for 68 runs.
Even after their own collapse, the efforts of Jones and Dean ensured that the work of England's bowlers was not in vain.