England are 61 runs away from securing their first victory under new head coach Brendon McCullum as his side made good progress chasing a challenging target of 277.
The hosts reached 216 for five at stumps, with Stokes going out for 54 and Root remaining on 77.
The fifth-wicket partnership dragged England from a desperate looking score of 69/4 as the two men combined to hit 90. Their time together at the crease was ended by Kyle Jamieson, who bagged his fourth wicket of the match.
Thing could've gone in a different direction when the new skipper fell to an ugly hack which briefly left the scoreboard reading 76 for five.
But his dismissal for just one run was scrubbed from the records when replays showed Colin de Grandhomme overstepping, a no-ball that changed the shape of the game.
Before England went to bat, they first had to dispatch New Zealand, and they certainly made a good start by taking three wickets in three balls to spark a collapse of six for 34.
Daryl Mitchell scored a century, but after he was out for 108 and Tom Blundell for 96, the tourists were eventually dismissed for 285.
Stuart Broad led England's bowling attack as the veteran bowler took three wickets, including the crucial one of Mitchell.
That was followed by the ups and downs of the England chase, leaving the prospect of a grandstand finish on the morning of the fourth day.