England were 337 for seven at stumps on the second day after Root made 104, having come in with England in trouble on 67 for three following yesterday's first day wash-out.
Together with Yorkshire colleague Jonny Bairstow (64), Root shared a fifth-wicket stand of 124.
However, left-arm quick Trent Boult brought New Zealand back into the match with a new-ball burst of three wickets for three runs in 13 deliveries to remove both Yorkshiremen and Stuart Broad.
The 22-year-old Root, in his sixth match at this level, was the first Yorkshire batsman in the county's 150-year history to make his first Test hundred at Headingley.
It was another composed display by Root following his 71 in England's 170-run first win in the first of this two-Test series at Lord's last week and means he has now scored 861 first-class runs in seven innings this season.
But Tim Southee, who took 10 wickets at Lord's, had a static Nick Compton excellently caught head-high by third slip Dean Brownlie for one.
England then made largely untroubled progress during a second-wicket stand of 56 between Cook (34) and Jonathan Trott (28).
But Trott, driving well away from his body, was caught behind by New Zealand captain Brendon McCullum off left-arm seamer Neil Wagner.
And next ball left-hander Cook flat-footedly edged recalled paceman Doug Bracewell, part of a four-man seam attack with spinners Bruce Martin (injured) and Daniel Vettori (unfit), to Brownlie as 67 for two became 67 for three.
Conditions became tougher for New Zealand's seamers but occasional off-spinner Kane Williamson lived up to his reputation as a partnership breaker when he had Bell, carelessly caught behind for 30 by McCullum, keeping wicket after BJ Watling was ruled out with a knee injury he suffered at Lord's.
It was the end of a stand worth 79 and, at tea England were 178 for four with Root 61 not out and Bairstow 13 not out.
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