Root, who received on-field treatment, was bounced again on the next delivery and caught off the glove by BJ Watling to leave England in desperate trouble at 132 for three, needing another 237 runs to make New Zealand bat again.
Until those last two deliveries on Sunday, Root had looked like he might be able to defy New Zealand on the final day, having reached his 38th test half century before Boult removed him for 51.
Dawid Malan was 19 not out and will be joined by Ben Stokes, who will face the final ball of Boult's 14th over when play resumes on Monday.
Earlier, New Zealand captain Kane Williamson had declared his side's first innings at 427-8, a lead of 369 runs, about an hour into the middle session. England were dismissed for just 58 in their first innings on Saturday.
Rain had washed out virtually all of the previous two days, with only 23.1 overs bowled by England on Friday and just 17 deliveries on Saturday.
Finally able to enjoy a period of uninterrupted play, Henry Nicholls advanced from his overnight score of 52 to reach his highest test score of 145 not out before Williamson declared.
Boult then made an early breakthrough in England's second innings when he had a woefully out-of-form Alastair Cook caught down the leg side by wicketkeeper Watling for two.
Mark Stoneman and Root had looked comfortable after seeing off the threat of Boult and fellow opening bowler Tim Southee as they compiled an 88-run partnership and appeared well set to go through to the close.
Stoneman, however, undid all of their good work when he dispatched successive Neil Wagner deliveries for a four and then a six that brought up his fourth test half century, only to be caught by Boult in the deep off the next ball.
Root and Malan then continued to defy New Zealand's bowlers, with the 27-year-old captain bringing up his half century with a cut past point for two runs, before the dramatic final two deliveries put New Zealand firmly back in control.
The second test, a standard daytime match, is at Hagley Oval in Christchurch from March 30-April 3.
(Reporting by Greg Stutchbury in Wellington; Editing by John O'Brien)
