Wagner's final figures of 4 for 62 in 15.2 overs included both of England's set batters, Ben Stokes and Joe Root, as well as the crucial final scalp of James Anderson, caught down the leg side by Tom Blundell, as England slipped from 201 for 5 to 256 all out in the course of his final gut-busting ten-over spell.
And the performance capped an extraordinary comeback from one of New Zealand's most indefatigable performers, after he had borne the brunt of England's aggression, both in second innings at Mount Maunganui, where his figures of 13-0-110-2 had been the second least economical in Test history, and in the first innings at Wellington, where Harry Brook's thrilling 186 had dispatched him at close to a run a ball across 21 more overs.
This time, however, with England reeling in their run-chase after the loss of four early wickets on the final day, Wagner's aggression proved the difference, with both Stokes and Root falling in consecutive overs in failed attempts to capitalise on a deck-hitting approach that proved so effective for New Zealand throughout their reign as World Test Champions.
"I got a bit of rhythm, something ticked which is nice," Wagner said in the moment of victory. "I guess it happens in cricket. But credit to Harry Brook, he's a serious talent. The way he's played it and came after me, he was pretty awesome to watch but not to receive. He's a serious player but to finally get some reward from it was quite pleasing."
"That's the characteristics of this team, we keep having to fight for each other, find a way of doing the hard yards out there, and we did," Wagner said. "It's a special one, this, and we'll celebrate it well. It's an amazing achievement, and obviously everybody contributed, so hats off to everyone. That's what this team is about, to keep fighting and it's just something that we're extremely proud of."
"It's hard to go past having only been the fourth team to win a Test match when asked to follow-on," he said. "I imagine it'll be a Test match that's talked about for a long time.
"We went back to what Wags' biggest strength is," Southee added. "He bought into it, he trusted it, even though it hadn't come off as he'd have liked in the series. We trusted his best method and he was able to come in and change the game in this last session, like he has done for a long period of time.
"It's an unusual tactic that you see a little bit more of nowadays, but it's something Neil's done for a long period of time. For him to come in and change the game there, when it looked like Ben and Joe had almost taken it away from us was a massive part of this game, and shows you the ticker that Neil's got.
"It shows more about him as a character and a cricketer. He doesn't give up. It's in his DNA to keep giving to this team. And I think we saw that and how valuable he can be: when nothing was really happening, and into the wind as well, was able to change the game in that last sessions."
McCullum's own reign as New Zealand captain, from 2013 to 2016, was instrumental in instilling the fighting spirit that endures to this day, and he paid special tribute to Wagner, a man whom he first played alongside in the Caribbean in 2012.
"It's a tough game, right, and tough characters have to find a way and they do," McCullum said. "Neil Wagner is one of the toughest that I've come across. Obviously I had the pleasure of captaining them for a long period of time, and now playing against him, you know that he's got a huge heart and he'll find a way when the going gets tough.
"He was good today. He was better than good, he was excellent. He turned the game on its head."
Andrew Miller is UK editor of ESPNcricinfo. @miller_cricket