Hazlewood's rant, after an lbw appeal against Kane Williamson was turned down by both the on-field and third umpires, was picked up by a stump microphone.
Williamson hit 97, while BJ Watling and Matt Henry added 118 for the eighth wicket as New Zealand scored 335.
Chasing 201 for a 2-0 series win in Christchurch, Australia closed on 70-1.
David Warner was the batsman out for the tourists, although that too was not without a little controversy.
An appeal for caught behind was turned down by umpire Ranmore Martinesz, but the decision was overturned on review by third umpire Richard Illingworth after television replays proved Warner gloved the ball down the leg-side to keeper BJ Watling.
However, Martinesz appeared to again give Warner (22) not out before Illingworth intervened, saying three times "give him out", at which point the Sri Lankan official raised his finger.
When the pair earlier refused to dismiss Williamson Hazlewood became angry and used offensive language, despite television technology showing the New Zealand opener edged the ball into his foot, meaning he could not be out lbw.
Williamson was out nine runs later, bowled by Jackson Bird, who then had Tim Southee caught by Smith for a second-ball duck to leave the hosts on 210-7, a lead of just 75.
Watling (46) and Henry (66) went on the attack, with the latter hitting 12 boundaries as he easily bettered his previous best Test score of 27.
But he was bowled by Bird, seven runs after Watling had been dismissed by James Pattinson (4-77), with the Black Caps on 335.
Bird then had Trent Boult caught by Pattinson for a duck to finish with 5-59 - his first five-wicket haul in Tests.
In reply, Australia - who will regain the top spot in the ICC Test rankings with a draw - were 70-1 at stumps with Joe Burns 27 not out and Usman Khawaja unbeaten on 19.
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Australian pace bowler Josh Hazlewood has pleaded guilty to dissent after swearing during day four of the second Test against New Zealand.