The hosts declared on their overnight total of 264-4, but rain forced an early lunch with the Kiwis 48-1.
Opener Martin Guptill resisted for 133 balls for his 23, while Kane Williamson added 59 to his first-innings 140 but fell just before tea.
The rain returned to wash out the final session with the Black Caps on 142-3.
With only 53 overs possible on day four, the fifth and final day will start half-an-hour earlier at 23:30 GMT.
When play does resume, much will depend on the two experienced men at the crease - New Zealand captain Brendon McCullum (four not out) and his predecessor Ross Taylor (20 not out) - if the tourists are to save the Test, although the weather may again play a part.
Guptill was dropped at short leg in the second over but Tom Latham was the first wicket to fall when he was trapped lbw by an inswinging yorker from left-arm paceman Mitchell Starc.
But it was off-spinner Nathan Lyon who took the other two wickets, finally inducing an edge to slip from the obdurate Guptill.
Williamson was adjudged lbw but he narrowly failed to overturn the decision on review, the ball-tracker suggesting the ball may have just clipped the top of the bails with an "umpire's call" verdict from the third umpire upholding the on-field decision.
Listen to ball-by-ball commentary of every day of the Test series on BBC Radio 5 live sports extra from 23:30 GMT.
Rain in Brisbane held up Australia's pursuit of victory on day four after they set New Zealand a world record target of 504 to win the first Test.