Having been set 216 to win, Australia skipper Mitchell Marsh hammered 72 off 44 balls before batting partner David (31 not out) sealed a thrilling last-gasp victory.
"I thought we were done, to be honest," admitted the relieved Marsh.
"Timmy David is extremely calm and had a lot of confidence in his ability towards the end. We are very lucky to have him. It was a great game of cricket."
The three-match series is the last for both teams before the T20 World Cup in the United States and West Indies in June.
Rachin Ravindra had earlier put New Zealand in the driving seat by hitting 68 runs off 35 balls to claim his first half-century in the format as the hosts finished on 215-3.
Marsh finished with best bowling figures of 1-21 before leading the charge with the bat, hitting seven sixes and two fours in his unbeaten knock.
The captain steadied his side after they lost openers Travis Head and David Warner early on.
Head went for 24, caught by Tim Southee off the bowling of Adam Milne. Veteran Warner followed for 32 in the sixth over, caught by Glenn Phillips.
Having criticised New Zealand fans in the build-up, Warner blew kisses at the crowd as he was booed off the pitch.
Marsh built a partnership of 42 for the third wicket with Glenn Maxwell, who was clean-bowled by Lockie Ferguson on 25 to leave Australia 111-3 after 10 overs.
New Zealand's fielding let them down in the 12th over as Southee missed a catch off Josh Inglis, then Phillips dropped Marsh's next ball on the boundary.
Inglis was dismissed with three overs left with Australia on 172-4 when he skied his shot to Ravindra at deep backward square leg. Inglis made 20.
The game went to the wire after New Zealand seamer Milne was smashed for 19 runs in the penultimate over.
Australia needed 15 from the last six balls, but David held his nerve by smashing a six before sending the final ball of the night to the boundary past a diving Phillips.
"I thought we didn't bowl too bad, but the power they possess all the way down is the reason why they are pretty good," New Zealand captain Mitchell Santner said of Australia.
"We had a pretty good score, but we know they keep coming with the bat. The only way to stop them is to take wickets and we didn't really do that."
The series moves to Auckland with the next two T20 internationals both to be played at Eden Park, on Friday and Sunday, before a two-Test series.
