New Zealand resumed on 44-2 in reply to Australia's first innings of 467 and reached 102-6 at lunch, losing four wickets for 58 runs to trail by 365 on the first innings. Cummins boasts the excellent figures of 3-17 from 12 overs.
Ross Taylor, Henry Nicholls and B.J. Watling all failed to reach double figures, before Colin de Grandhomme fell for 11.
Mitch Santner appeared to be out caught at gully off the glove off the bowling of Mitchell Starc, but was given not out and a video review failed to overturn the decision.
Opener Tom Latham, dropped in slips on nine, was unbeaten on 42 at lunch with Santner on one.
Cummins was on a hat-trick in the third over of the day after removing Taylor (4) caught at first slip and Henry Nicholls (first-ball duck) lbw in consecutive deliveries as New Zealand slumped to 46-4.
Ex-captain Taylor edged to Joe Burns who took the catch at first slip, after the ball had burst through the hands of Marnus Labuschagne at third slip. Nicholls unsuccessfully tried to have his decision overturned by calling for a video review.
Pattinson had Latham dropped at second slip at 51-4. Pattinson also troubled Watling (seven), who fended at a short ball and was caught at first slip at 58-5.
In similar fashion, de Grandhomme edged a rising delivery from Mitchell Starc to gully at 97-6.
On Friday, Travis Head scored a crucial century in Australia's first innings of 467 before New Zealand lost 44-2. Head shared a 150-run partnership with skipper Tim Paine to tilt the game firmly in Australia's favor.