After being set 187 runs to win, the hosts were rocked by the double loss of skipper Steve Smith and his deputy David Warner in the space of seven balls.
But the West Australian pair restored the innings with an unbeaten 47-run fourth-wicket stand on the third day.
At the dinner break heading into the crucial final session under lights, Australia were 113 for three and 74 runs from victory with Voges on 27 and Marsh not out 24.
Whichever way it goes, the Adelaide Oval is poised for its first three-day Test match in 64 years -- since the West Indies beat Australia by six wickets.
More From This Section
The first wicket came in the sixth over when Trent Boult trapped Joe Burns leg before, but Steve Smith was dropped five overs later.
Smith pulled Doug Bracewell into the air towards Mitchell Santner on the leg-side, but the Kiwi debutant put down the skied chance to give the Australia 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's departure brought Shaun Marsh to the crease at a crucial stage of the Australian innings as he fought to save his stop-start Test career.
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 runs with Santner top-scoring with an impressive 45 off 88 balls.
B.J. Watling fell on the ninth ball of the day on his overnight score of seven.
Hazlewood induced an edge which flew to Smith at second slip, who had dropped Watling on two the previous evening off the same bowler.
After jumping down the wicket to hoick spinner Nathan Lyon for six, the bowler two balls later lured Santner to repeat the shot -- and he was smartly stumped by wicketkeeper Peter Nevill.
Southee then hit two fours and went after Mitch Marsh but was well caught by Lyon on the backward square boundary for 16. Boult went for a wild swing and was bowled by Hazlewood for five to end the innings, with Bracewell unbeaten on 27.