Martin Guptill (42 off 50) and Grant Elliott (36 off 38) also offered solidity to the order before Jimmy Neesham (37 off 31) and Nathan McCullum (25 off 16) delivered pep in the death overs.
Tom Latham was victim to a sharp Graeme Cremer catch at short mid-wicket for 16.
Cremer's leg spin also came to the fore, taking three for 44 from his allotment.
Colin Munro replaced Ross Taylor in the starting XI after the resident No.4 was invalided out of the tour with a groin injury. Munro's struggle to find form at international level continued. Cremer bowled him for nine off 14 balls as he swung across the line.
The form of Luke Ronchi (seven off 15) is also due scrutiny. Since the start of the World Cup he has passed 13 twice in 13 ODI innings.
Despite a world ranking of 11, Zimbabwe remained competitive for most of their innings.
New Zealand restricted them to 37 after the initial powerplay, but Hamilton Masakadza (57 off 71) and Chamu Chibhabha (32 off 65) delivered an opening stand of 97. Further thwarting momentum was the loss of series century-makers Craig Ervine for 32 and Sikandar Raza for five. New Zealand responded to the opening partnership by taking six for 80 in 15.2 overs, although Sean Williams (63 from 62 balls) ensured a rollicking chase.
Mitchell McClenaghan (three for 36 from 8.4 overs) and Ish Sodhi (two for 47 from 10) were the pick of the bowlers although Ben Wheeler (none for 34 from nine) built a frugal platform. He conceded just 14 runs from his five-over spell.
"The wicket was tough after getting put in by Zimbabwe who bowled well," Williamson said. "It wasn't easy, and we initially thought the total was good until they came back and put us under pressure."