New Zealand held a lead of 43 runs at tea after rolling Australia for 136 in the first innings. After the tea break, the Kiwis lost openers Brendon McCullum and Martin Guptill in quick succession as the Aussies fought back after a poor batting display.
McCullum was the first to fall, edging James Pattinson to Phil Hughes at third slip while Guptill fell to Peter Siddle at the same score, when he feathered a catch to 'keeper Brad Haddin.
The burly Jesse Ryder tried to make amends for his first innings duck and looked to be finding his touch when he was stumped brilliantly by Haddin of Mike Hussey's bowling, for 16.
The skipper (Taylor) and the talented Kane Williamson pushed the total past 100 as they mixed caution with aggression.
Fast bowlers Peter Siddle and James Pattinson helped the Australians avoid their lowest Test score against the Kiwis with a 56-run eighth-wicket stand after the home side went to lunch at 81 for seven.
At tea, New Zealand had added to their 14-run innings lead to be 29 without loss with Martin Guptill on 12 and Brendon McCullum on nine.
Australia struggled in their pursuit of the Kiwis' first innings of 150 and at one stage were on track for their lowest Test total against New Zealand of 103 set in Auckland in 1986.
But Siddle (36) and Pattinson (17) saved their blushes with their partnership as the Bellerive pitch began to play easier after its first-day demons.
New Zealand's all pace attack kept the Australians under pressure with Chris Martin (3 for 46), Trent Boult (3 for 29) and Doug Bracewell (3 for 20) among the wickets.
Australia lost six wickets in the extended morning session after resuming at 12 for one following Friday's loss of Phil Hughes for four.
David Warner only lasted 17 minutes into the second day before he nicked Martin to Ross Taylor at first slip for 15.
Ricky Ponting walked before he was given out leg before wicket for five after playing Tim Southee across his pads.
It was the fifth time in his last six Test innings that Test cricket's third-greatest runscorer was out lbw and the 30th Test innings since his last century.
Ponting's last big knock was 209 against Pakistan almost two years ago in Hobart.
Usman Khawaja's watchful 51-ball vigil ended when he edged Martin to wicketkeeper Reece Young for seven in 77 minutes and left Australia 35 for four.
Mike Hussey tried to get his bat out of the way of a Boult lifter only to bottom-edge a catch to Young for eight to send Australia reeling at 58 for five.
Brad Haddin was out to a loose shot finding Brendan McCullum at mid-off for five and giving Bracewell his first wicket.
Michael Clarke did not offer a shot and was surprised by Bracewell's low bounce to be bowled for 22.
But Siddle and Pattinson eased Australia past their trans-Tasman Test lowest score before Siddle was caught in the slips by Guptill off Bracewell and Pattinson fell in the next over, brilliantly caught in the gully by Kane Williamson off Boult.
Boult took his third wicket in his debut Test trapping Mitchell Starc lbw for four.