Off-spinner Swann took six for 90, for a Test-best match return of 10 for 132, as New Zealand were dismissed for 220.
This win gave England a 2-0 win in the two-match series after their 170-run victory in the first Test at Lord's.
Swann's figures topped his previous Test match-best of 10 for 181 against Sri Lanka in Colombo last year.
The scale of New Zealand's task was that their victory target of 468 was 50 more than the Test record fourth innings winning total of 418 for seven made by the West Indies against Australia in Antigua in 2002/03.
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"I never thought I'd do it at Headingley. I actually agreed with the decision not to pick me here last year as I bowled like a drain at The Oval and it looked a seamers' pitch," Swann told BBC Radio.
This was England's last Test before they begin the defence of the Ashes against Australia in July and Swann added: "I know we're not allowed to say the 'A' word but it was great to get some wickets ahead of the Ashes."
"We all know how good a bowler he is, he doesn't need the help from the footmarks but when he gets it he's nigh-on unplayable."
Rain, as much as New Zealand's lower-order batting, was the biggest obstacle to England recording a thumping win on the final day.