It was a first test victory for England in New Zealand since 2008 and a 10th win in 11 matches since former Black Caps skipper Brendon "Baz" McCullum and Christchurch-born Ben Stokes took over as coach and captain.
The Black Caps were perhaps still shellshocked after Stuart Broad's brilliant spell with the pink ball on Saturday evening and only Daryl Mitchell offered any real resistance as they slumped to 126 all out in the opening session of the fourth day.
"Another great performance," said Stokes. "Very clinical with the bat and very clinical with the ball. We executed as well as we wanted to."
"It was entertaining cricket and that's what we want to do. We're just trying to put on a show."
Broad (4-49) laboured in vain on Sunday to get his 20th five-wicket haul in tests, but his strike partner James Anderson (4-18) stepped up to lead the mopping up of the New Zealand tail.
See the scorecard of the first test at Flashscore
The pair on Saturday supplanted Australia's Glenn McGrath and Shane Warne as the most prolific test bowling partnership and now have a combined 1,009 wickets from 113 matches together.
"Having James Anderson and Stuart Broad in your side ... does make captaincy a lot easier, because you just throw them the ball and they'll probably get a wicket for you," Stokes said.
"But not only were they brilliant this whole game with the ball but ... we've got a 40-year-old and a 36-year-old setting the standard of what we're about in the field in terms of energy."
"They're setting a great example for anyone who wants to look at having a long career in professional sport," Stokes added.
"Seeing them with three Lions on their chests is great and I don't really want to think too far ahead as to when they might call time on their careers."
Stokes made a bit of cricketing history himself during the match, hoisting a Scott Kuggeleijn delivery over the fine leg boundary to supplant coach McCullum as the most prolific scorer of sixes in test cricket with his 108th maximum.
"I will say that I did know that I was on that one for a long time," he said.
"I was blowing that one over because I didn't know if it was gonna go or not. But when I saw it go over, it was just like I just picked Brendon up and dropped him on the floor."
World test champions New Zealand had resumed on 63-5 with the mammoth task of chasing down 394 for victory, but England quickly resumed their dominance of the previous evening.
Michael Bracewell (25), Scott Kuggeleijn, Tim Southee and Neil Wagner departed so swiftly that it looked like the victory would be wrapped up inside an hour.
Anderson had dismissed Kuggeleijn for two and New Zealand skipper Southee for a golden duck with successive deliveries but Mitchell fended away the hat-trick ball and proceeded to stall the rout for nearly an hour with an unbeaten 57.
ENGLAND STRATEGY
England's strategy of batting aggressively during the day and doing their best to expose New Zealand's top order to the vagaries of the evening sessions paid huge dividends.
Stokes declared on 325-9 after England were sent in to bat on day one and although New Zealand responded with a first-innings tally of 306, the tourists always appeared to be in control of the game.
After another "Bazball" onslaught at the start of day three, the England batsmen slowed the pace to extend the innings and ensure New Zealand started their second knock under the floodlights.
England's 374 gave them a healthy lead but they barely needed it as Broad bowled Devon Conway, Kane Williamson and Tom Latham through the gate in short order before returning to remove first innings centurion Tom Blundell in the same style.
Promising young England batsman Harry Brook was named Man of the Match for his half centuries in each innings.
"Brooky's just carried on from his amazing series in Pakistan," Stokes said. "He's a fantastic talent and I think he's going to be a global superstar."
New Zealand have lost or drawn their last five test series and now need to win the second test in Wellington, which will start on Friday, to save this series.
"Disappointing, but I think credit to England. They played it strategically very well," said Southee.
"We were on the receiving end of a couple of night periods that weren't obviously ideal. We've got a couple of days to digest things and look forward to getting back to red ball cricket at the Basin Reserve.
"It was not long ago that Kane got a double hundred and Devon got a hundred. But I think they had the worst of the conditions both times having to go out under lights, but if you look at our top three and they're three of the best players we've had.
"So as a group we know and we trust that they'll come right when we go back to Wellington with a more traditional test match."
Southee, a former teammate of McCullum's, said he thought England's entertaining style was good for the game but that the aggressive batting did create chances for bowlers.
"I think it's very exciting," the seamer said.
"As a bowling side you're excited about coming up against it because it presents opportunities as well, that there's only one way they're going to play and one way they know how to play."
"If we hang tough, and are a little bit more disciplined, then things can go our way."
Southee said he hoped Matt Henry, who missed the Bay Oval test to remain in Christchurch for the birth of his first child, would be back to bolster the pace attack when the second test started on Friday.
The 34-year-old said there was not much doom and gloom in the dressing room as New Zealanders tended "not to ride the emotions too much", even when things were going well.
"We're still enjoying ourselves, enjoying each other's company and trusting each other and what we're trying to do," he said.