BJ Watling scored the first Test century on Bay Oval as New Zealand established a 41-run lead against England on Saturday.
He walked off at the end of the third evening with 119 runs from 298 balls, to set New Zealand 394 for 6.
Only two wickets fell all day, with Joe Root dismissing Henry Nicholls for lbw in the morning session.
Colin de Grandhomme supported Watling in an excellent 39-over stand, until he was caught by Dom Sibley for 65 off Ben Stokes.
In the evening session, he Mitchell Santner, batting on 31, for company.