New Zealand, chasing 239 for victory, were dismissed for just 68 in under two hours, with Broad taking seven wickets for 44 runs in a mere 11 overs.
South Africa born duo BJ Watling (13) and Neil Wagner (17), who got to 10 with a six off Broad, were the only New Zealand batsmen to reach double figures in an innings that lasted less than 23 overs.
Man-of-the-match Broad praised the contribution of new-ball partner James Anderson, who took five first innings wickets on his way to becoming only the fourth England bowler to reach the landmark of 300 Test wickets.
"It's a great way to start the summer...It was an exciting game to play in and it was great to contribute to a Test win," the fast-medium bowler added after surpassing his previous Test-best of seven for 72 against the West Indies at Lord's last year.
England captain Alastair Cook, trying to explain the dramatic finale, told BBC Radio's Test Match Special: "Cricket's a strange game and sometimes results like that happen. I think the extra heat today helped the ball swing, it was a good wicket but it kept doing enough."
"It's a tough one to explain, I thought we had parity or dominated for long periods but we took a step backwards today," said captain Brendon McCullum, whose side were dismissed for 43 by South Africa at Cape Town in January,
"The margin flatters England, but they bowled superbly in that hour before lunch which ripped the heart out of us.
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