Root's determined 68 helped England set New Zealand a testing target of 192, which proved too much for the tourists who collapsed from 89 for three to 135 all out in 16.3 overs at Old Trafford in Manchester.
New Zealand made a flying start with Kane Williamson hitting 57 from 37 balls, but wickets fell regularly and when he departed at 131 for seven, England swiftly wrapped up the tail to seal victory.
The result continued England's resurgence in white-ball cricket after they secured a superb 3-2 victory in a pulsating and run-filled one-day series on Saturday.
A fluctuating test series between the sides was drawn 1-1.
"Now we have won this game and the ODI series it reinforces to the guys that, it might not always come off, but we can beat the best in the world in any game of cricket," England captain Eoin Morgan told Sky Sports.
He had man-of-the-match Root to thank for setting up England's victory after the right-hander came to the crease with his side on 25 for one.
Root scored 68 runs off 46 balls, including eight fours and a six, before he mistimed a slower delivery from Mitchell McClenaghan to Martin Guptill in the deep having steered England to 157 for six.
Late hitting from Ben Stokes helped England set the tourists an imposing total, but New Zealand came out with all guns blazing.
Captain Brendon McCullum smashed 35 from 15 balls, including four sixes, before a yorker from Mark Wood forced him to edge the ball on to his stumps.
Williamson took over the batting mantle, but lacked support as England turned the screw.
Williamson departed when he was run out by a direct hit from David Willey and his exit ended New Zealand's hopes of victory.
England wasted little time in mopping up the tail to leave them in a very positive frame of mind going into the Ashes against Australia starting next month.
(Reporting by Toby Davis, editing by Ed Osmond)
