England had looked well set to bat out the final two sessions having resumed after lunch on 98-4 with Joe Denly on 21 and Ben Stokes on 13.
The pair safely negotiated the first 30 minutes but Stokes then played on for 28, letting out a loud yell of exasperation after chasing a wide, shortish ball from Tim Southee that he did not need to play at. His dismissal at 121-5 sparked a collapse, with Wagner at the heart of it.
Denly fell for 35 when he was caught by wicketkeeper BJ Watling following a lifting delivery from the left armer.
Ollie Pope also fell to Wagner, needlessly throwing his hands at a wide full toss only to watch the ball fly to a diving Mitchell Santner at short cover.
With Trent Boult off the field with sore ribs, Wagner took the second new ball and bowled Jos Buttler for an 18-ball duck with a yorker with the first delivery.
Joe Root was the only wicket to fall in the first session as his side dug in, with the England captain steering a lifting Colin de Grandhomme delivery to Tom Latham in the gully for 11.
Santner had reduced England to 55-3 at the close of play on Sunday after taking three quick wickets in the final 30 minutes of play.