Mitchell Santner was awarded man of the match after picking up seven wickets, removing the final three England batters not long after lunch on day four.
England ended up on 234/9 after captain Ben Stokes didn't bat in the final innings after pulling his hamstring during a long day in the field yesterday.
Southee picked up two wickets in his final innings as a Black Cap, opener Ben Duckett before the close of play last night and Jacob Bethell to finish with figures of 2/34.
England, who had already sealed the series by winning the first two tests, crumbled after lunch in their improbable chase of 658 runs to sweep the series.
Mitchell Santner took 4-85 after ripping through the England tail to give New Zealand their biggest ever win by runs and inflict the fourth-worst defeat by runs on the tourists.
The end came swiftly as England had looked determined to go down fighting before the break and continued going for their shots after it.
Gus Atkinson blasted 43 off 41 with seven fours and one six and his fellow seamer Brydon Carse clubbed 11 runs off 12 deliveries.
Jacob Bethell had earlier continued his impressive start to test cricket with a third half-century of the series before he holed out for 76.
Joe Root reached his 65th test half-century before being dismissed for 54 by Santner, whose appeal for lbw was initially turned down but upheld on review.
Harry Brook, whose runs were instrumental to England's wins in Christchurch and Wellington, faced six deliveries for one run before Will O'Rourke had him caught in the slips with the sort of steepling delivery that is fast becoming his trademark.
The big New Zealand fast bowler reached speeds of 153kph during a hostile spell of bowling that had the England batters jumping.
England's Harry Brook was awarded player of the series, despite only managing a solitary run in the Hamilton test.
- Reuters/APL
