New Zealand got off to a positive start with Tim Southee accounting for Dawid Malan early on in the day.
Jonny Bairstow lived a charmed life but fell to Tod Astle for 26.
Moeen Ali fell to Trent Boult as England went to Tea at 217 for 6.
Ben Stokes made 66 off 188 to frustrate New Zealand in the second session.
Chris Woakes and Stokes scored 83 for the seventh wicket before the latter fell at the dinner break leaving England at 300 for 7.
Woakes scored 52 before he fell to Wagner in the final session.
Wagner picked up the crucial wickets of Stoneman, Stokes and Woakes to end with 3 for 77.
Todd Astle picked up three wickets, and cleaned up the tail, to help New Zealand seal the Test by an innings and 49 runs and take 1-0 lead in the two-match series.