All five Australian bowlers claimed wickets with Mitchell Starc taking three for 43 while his new-ball partner Josh Hazlewood took two for 18 from eight overs
Conway blasted 126 from 110 balls and Mitchell finished on 100 not out as New Zealand ended their innings at 318 for six after winning the toss and electing to bat.
"We've got a great record here. A lot of our guys, this is their home ground so they certainly love playing here, they love being at home, they enjoy the conditions," Paine said.
Fellow fast bowlers Sean Abbott and Michael Neser are also in the squad, with the series finely poised after Australia won the first Test in Adelaide and India the second in Melbourne.
It was a determined and crucial knock against a world-class attack by the mild-mannered 32-year-old, who assumed the captaincy from Virat Kohli after he returned home for the birth of his first child.
It proved slow going as he and Mayank Agarwal rode their luck, with Pujara surviving again when a thick edge off Starc flew past Nathan Lyon's outstretched arm.