Shaun Marsh took Australia to the cusp of victory before losing his wicket on 49 and the fighting Kiwis set up a tense finish under the floodlights by also dismissing Peter Nevill to leave the home side seven wickets down.
Peter Siddle hit the winning runs, much to the relief of the crowd, to clinch a 2-0 series win over the gallant Black Caps.
Australia won the first Brisbane Test by 208 runs and the second Perth Test was drawn.
The pink ball developed for day-night Tests triumphed over the bat in a fast-paced Test which attracted a total attendance of 123,736. The aim of the experiment is to bring the crowds back to the long form of the game.
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"The bowlers have dominated this game and it's great for cricket," said Australian skipper Steve Smith.
"It has certainly brought the bowlers into the game, particularly under the lights."
It was the first three-day Test match at the Adelaide Oval in 64 years -- since the West Indies beat Australia by six wickets.
"In terms of an experiment I thought it went off as good as it could do, and the crowd supported it and it was an outstanding game of cricket."
Shaun Marsh had been dropped during the Ashes series in England in August and only came into the Adelaide Test as a replacement for injured number three Usman Khawaja.
He lost his younger brother Mitch during the final session, caught holing out to spinner Mitchell Santner for 28 with 26 runs still needed for victory.
Adam Voges, who had earlier shared a valuable 49-run fourth-wicket stand with Marsh, was out early in the final session -- snapped up at second slip by Tim Southee for 28 off Boult.
Boult claimed the first wicket of the innings in the sixth over by trapping Joe Burns leg before, but Smith was dropped five overs later.
Smith pulled Doug Bracewell high into the air towards Santner on the leg-side, but the Kiwi debutant put down the skied chance to give the Australian skipper a massive letoff on two.
And the Kiwis got another major fillip when Smith was out leg before wicket to Boult seven balls later.
Smith reviewed the decision and the ball was tracked hitting him high on his pad. But the Hawk Eye technology had the ball just flicking the leg bail and that was enough to back up umpire Richard Illingworth's on-field decision.
Smith left the field shaking his head after scoring just 14 and leaving his team on a precarious 66 for three, still 121 runs away from victory.
New Zealand earlier lost their last five wickets in the day's first session to be bowled out for 208, a lead of 186.
Man-of-the-match paceman Josh Hazlewood led the Australian attack in the absence of the injured Mitchell Starc, taking six for 70 off 24.5 overs.
The Kiwis, who had resumed on 116 for five, lost their last five wickets for 92. Santner top-scored with an impressive 45 off 88 balls.