England captain Root ended his run drought with an unbeaten 114 while Burns survived on luck to post 101 but when rain ended play 16 overs early the tourists were 269 for five.
They trailed New Zealand by 106 with five wickets remaining and their vision of building a healthy lead rapidly fading. New Zealand struck crucial blows late on day three of the second Test to have England in trouble despite centuries from Joe Root and Rory Burns in Hamilton on Sunday.
With Root and Burns putting on 177 for the third wicket, England found the "bat-long" partnership they had been searching for in their attempt to follow the same recipe New Zealand used to win the first Test.
But from resuming the day at 39 for two and taking the score to 201, the dismissal of Burns triggered a slump that saw Ben Stokes gone for 26 while Zak Crawley made one on debut.
Root was under added pressure to not only guide England to a position of strength but also to end his own run dearth amid suggestions the pressure of captaincy was affecting his form.
To that end, he was in no mood to take risks as he faced 258 deliveries in six hours to reach 99 before his century came on a surprise mistiming that saw an inside edge fly past the wicketkeeper to the boundary.
His 17th Test hundred was his first in 15 innings since the West Indies in February.
Burns lead a charmed life on the way to his second century when he was dropped on 10 and 19 and on 87 survived a run-out opportunity when New Zealand botched the return throw.
But two balls after reaching three figures he ambled through the first single when Root called for two and although he accelerated on the return leg he was a centimetre short when BJ Watling broke the stumps.
Stokes strayed from the "bat-long" script when he replaced Burns and compiled a brisk 26 before Tim Southee extracted slight movement off the wicket and found an edge for Ross Taylor to hold.
Crawley's maiden Test innings only lasted six balls when he was caught behind to give Neil Wagner his first wicket.