Rain at Hagley Oval delayed the start of Australia's pursuit of the 202 runs they still needed for victory on Monday and New Zealand skipper Tim Southee struck early to dismiss Travis Head and reduce the tourists to 80-5.
Carey and Mitchell Marsh (80) both had scares but put together a partnership of 140 for the sixth wicket to subdue the crowd and take the visitors to within 60 runs of their target.
Ben Sears (4-90) revived New Zealand hopes of a first home win over their neighbours in 31 years when he removed Marsh and Mitchell Starc in successive balls, but Carey and skipper Pat Cummins (32) got Australia across the line on 281-7.
Southee, playing his 100th test, gave New Zealand the perfect start to the day when he removed Head with the eighth delivery of the opening session.
In retrospect, however, Rachin Ravindra fumbling a straightforward catch that would have sent back Marsh for 28 off the previous delivery was perhaps the more significant moment.
Wicketkeeper Carey, who took 10 catches behind the stumps over New Zealand's two innings, came out to join Marsh and they moved towards the target at a fair lick.
Carey kept his wicket by virtue of reviews when he was on 19 and 37, while Marsh was saved on 55 when the technology showed the tiniest bit of bat on a Glenn Phillips delivery that hit him on the toe.
Lunch came and went with the atmosphere flat at Hagley Oval but the crowd burst into life when seamer Sears was brought on and trapped Marsh lbw before having Starc caught by Will Young at square leg.
Cummins came in to fend away the hat-trick ball and Carey survived another scare when Phillips failed to reach a tough catch that would have given Sears a five-wicket haul on his test debut.
The New Zealand bowlers lacked nothing in effort but edges flashed wide of the cordon and anything on the stumps was blocked as Australia's target ticked down.
Carey was ultimately denied his second test century but will care little as he and his captain, who struck a four to the point boundary to seal the win, steered his country to a sixth win in seven tests over the Australasian summer.
The Black Caps remain with just one test win in 24 attempts against their closest rivals this century, while the victory will see the Australians move above them into second place in the World Test Championships standings behind India.
