Conway made 39 and Latham reached his fourth T20 half-century, helping New Zealand's cricketers pass South Africa's 9-136 with 22 balls remaining to take a 2-1 lead in the five-match series on Friday.
Batting conditions have been tricky and matches in the series have been low-scoring. South Africa won the first by seven wickets, chasing down New Zealand's 91, and the Black Caps won the second by 68 runs, bowling out the opposition for 107.
South Africa batted out 20 overs for the first time on Friday, helped by a lively unbeaten 26 from 20 balls by Nqobani Mokoena in his first international innings.
Latham and Conway began their run chase quietly, taking five runs from the first two overs and 12 from the first three. They then took 14 from the fourth over, bowled by Mokoena, and 21 from the fifth by Lutho Sipamla to set New Zealand on the way to a comfortable win.
Conway fell in the 11th over, caught in the deep by Rubin Hermann off Keshav Maharaj, and Tim Robinson was out for 17 in the 17th over when the scores were tied. Nick Kelly came out and hit a single for the winning run from the next ball.
"You look at the depth we've got at the moment ... with a few guys away," said New Zealand captain Mitchell Santner, who won the toss and sent South Africa in to bat.
"It was a simple blueprint tonight. The pitch looked like it was doing a little bit early. It looked tricky and it was good the way the two boys (Conway and Latham) could soak that up."
The pitch at Eden Park typically is quick and bouncy and it provided that again, but also held up at times.
New Zealand used six bowlers and all six took wickets, including the spinners who helped their team make inroads in the South Africa top order.
Pace bowler Lockie Ferguson took 1-9 from four overs and was named man of the match.
Santner (2-21 off three overs) found turn and, after coming on in the fifth over, dismissed Jason Smith (10) and Connor Esterhuizen (15).
The fourth match is in Wellington on Sunday.
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