The tourists were dismissed for 197 with about 90 minutes of play remaining in the match in Mt. Maunganui after Wagner broke a stubborn 59-run ninth-wicket partnership between Jofra Archer and Sam Curran.
The left-armer had taken three wickets in 17 balls for one run as England collapsed from 132-5 to 138-8 before tea.
He had Archer caught in the deep and then trapped Stuart Broad in front on the next delivery to end England’s resistance and finish with figures of 5-44 from 19.2 overs.
“It feels great to win a test match,” New Zealand captain Kane Williamson said at the post-match presentation. “England are an outstanding side and it took a tremendous amount of hard work to get past their total.”
The victory was New Zealand’s 11th in 104 tests against England. It was also their second largest after beating them by an innings and 132 runs in 1984.
New Zealand now have three wins in their last four Tests against England, with one draw, and their second in succession by an innings after winning the day-night test in Auckland by an innings and 49 runs in 2018.
Wicketkeeper BJ Watling was named man of the match after his maiden double century of 205 and hefty partnerships with Henry Nicholls, Colin de Grandhomme and Mitchell Santner helped the hosts to a first innings of 615-9 declared. “It felt really good,” Watling said. “The key was to get partnerships, and the boys stuck in there at the end.
“We tried to build as much of a lead as possible and hoped the wicket would do funny things on day five.”
After Santner had taken three quick England wickets late on Sunday, the tourists resumed on 55-3 and faced an uphill battle to bat the entire fifth day to stop New Zealand from taking a 1-0 lead into the second match of the two-Test series on Friday in Hamilton.
And while they showed signs they might be able to see the day out, tentative shot- and poor decision-making led to their downfall.
Captain Joe Root, Joe Denly, Ben Stokes and Ollie Pope were all dismissed playing at balls they did not need to, while Archer fell into a trap set by Wagner.
Stokes’s dismissal was particularly important with the aggressive all-rounder looking settled before he played on, letting out a loud yell of exasperation after chasing a wide, shortish ball from Tim Southee.
His dismissal at 121-5 sparked a collapse, with Wagner then ripping through the lower order.
Archer and Curran kept the hosts at bay for more than an hour either side of tea and were starting to look comfortable before Wagner struck again to wrap up the match.
Root said failing to kick on from a decent platform in the first innings had cost England.
“We missed an opportunity if we’re brutally honest with ourselves,” said Root of England’s collapse from 277-4 to 295-8 in their first innings before they were dismissed for 353.
“Could have done with a score over 450, and using that scoreboard pressure as another fielder.
“We did a lot of good stuff, we just need to do it for longer.”