Set an imposing 463 for victory, England stumbled to 40-3 after early losses from Emilio Gay, Jacob Bethell and Ben Duckett.
Gay chipped Kyle Jamieson to mid-wicket, Bethell's difficult run of form continued with a first-ball lbw, and Duckett fell attempting to hook Will O'Rourke to mid-on.
With England in trouble, Root arrived at the crease and immediately reached another landmark. The stand-in captain received a standing ovation after moving to two runs, taking him beyond 14,000 Test runs and making him only the second batter after Sachin Tendulkar to achieve the feat.
Root then steadied the innings alongside Harry Brook, whose counter-attacking approach transformed the mood of the match.
After tea, England raced along, scoring 43 runs in the first five overs of the session despite the opening over being a maiden. Brook struck a six over square leg and reached fifty from only 33 deliveries.
Both batters survived lbw reviews against Matt Henry because of inside edges before combining for a vital 97-run stand. Henry eventually broke through, removing Brook for 58 when he edged to slip.
Debutant James Rew added a composed 15 and eased fears of England being dismissed inside four days before Jamieson trapped him lbw.
Root, meanwhile, remained unbeaten on 75 as England closed on 182-5, still requiring a further 281 runs. New Zealand need only one more wicket to expose the tail and are favourites to square the three-match series at 1-1.
Earlier, New Zealand resumed on 252-3 but lost their final seven wickets for 110 runs. Jofra Archer dismissed Henry Nicholls for 121 and Glenn Phillips, while Matthew Fisher claimed three wickets, including Daryl Mitchell for 68. Behind the stumps, Rew bounced back from two dropped catches on Friday with a much-improved display.
Away from The Oval, another storyline was unfolding as Ben Stokes, absent while an investigation into a London nightclub incident continues, scored 95 for Durham.
It was his highest score in any format since making a century against India almost a year ago and strengthened hopes of a return for the decisive Test at Trent Bridge.
Jos Buttler has been retained in England's squad for the five-match T20I series against India while James Coles has earned his maiden international call up in an enlarged 17-man group.
Matt Henry says his job is to be effective and not "look like a hero" after bowling New Zealand to victory in the second Test against England at the Oval to level the three-match series at 1-1.
Ben Stokes and Gus Atkinson have been cleared to return to England's squad for the third and deciding Test against New Zealand, after both players received written warnings following an investigation into a breach of team protocol.
New Zealand claimed a dominant 253-run win in the second Test against England to take the series to 1-1 ahead of the deciding game in Nottingham.
