Conway was dropped on 1, should have been run out on 34, survived a strong lbw appeal and miscued shots all over Harare Sports Club on Friday, but they all landed safely.
He ended up hitting the winnings runs, though, pulling Trevor Gwandu to the midwicket boundary as New Zealand won with 6.1 overs to spare.
Put in, Zimbabwe made only 7-120. New Zealand replied with 2-122, Conway top-scoring with an unbeaten 59 off 40 balls, his first T20 half-century in 17 months.
"Got a bit lucky early and fortunate I got through that period," Conway said. "It was nice to spend some time in the middle."
New Zealand now have two wins from two games in the T20 tri-series with South Africa.
Zimbabwe openers Wessly Madhevere and Brian Bennett greeted the New Zealanders aggressively. But the run rate was falling from eight an over even before Bennett, 20 off 19, gave Matt Henry his first wicket at the end of the powerplay.
New Zealand could count on three spinners to Zimbabwe's one, and Mitchell Santner and Rachin Ravindra put the brakes on the home side.
The only other decent scorer was Madhevere, who backed away to hit Adam Milne to the off side and lost his off stump for 36 off 32 balls.
Henry took 3-26 and Santner and Ravindra a wicket each.
When New Zealand batted, Blessing Muzarabani dropped Conway in the first over but dismissed Tim Seifert in the second.
As Conway rode his luck, Ravindra was on the charge with three consecutive boundaries off Gwandu and a six over deep backward point. His 30 off 19 lifted New Zealand more than halfway to the target.
Conway didn't look comfortable until he was in the high 30s, but he and Daryl Mitchell, 26 off 19, steered New Zealand home.
Zimbabwe face South Africa next on Sunday.
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