New Zealand kept their hopes in the first Test alive as Glenn Phillips took five wickets to bowl Australia out and set a target of 369 on the third day in Wellington.

Nightwatchman Nathan Lyon was the unlikely top scorer for Australia, falling on 41 with his first Test 50 in sight.

Cameron Green continued his form with 34, but all-rounder Phillips spun his way to five wickets and was on a hat-trick after removing Travis Head for 29 and dismissing Mitch Marsh with the next ball.

Phillips finished with 5-45 as Australia were all out for 164.

The Australians struck early in the Blackcaps innings, Lyon dismissing Tom Latham and Kane Williamson for single-digit scores, before part-timer Head removed Will Young for 15.

Rachin Ravindra (56 not out) and Daryl Mitchell (12 not out) dug in for the rest of the day.

Two days remain with New Zealand requiring 257 pull off what would be an incredible victory.