England head into day three at 2-50, still 393 runs behind New Zealand and with two of their best batsmen, Alastair Cook and Jonathan Trott, already dismissed.
Nick Compton (12), who has two centuries on this tour, and the experienced Ian Bell (6) will resume today, but Kevin Pietersen's absence through injury means Bairstow and Root, who have eight Tests between them, may play critical roles.
Root played well in the limited overs matches in New Zealand but has failed to date in the Tests, while Bairstow hasn't batted since the second Test in India in December.
English paceman Steve Finn says the experienced players will want to step up but that the team has faith in Bairstow and Root.
"Those guys have played some very important innings for England over the last 12 months," he said at the end of Saturday's play.
"It's great for them to have an opportunity to go out there and test themselves in this environment, because there's no better place to learn than out there in the test arena. We're really looking forward to tomorrow."
Trent Boult dismissed both Cook and Trott and was getting some good swing late on day two, while left-arm spinner Bruce Martin troubled the English batters with his flight, turn and bounce.
"If we can get the ball swinging there is a little bit of bounce there and if you hit the wicket you'll get as much as you can out of it," New Zealand batsman Kane Williamson said.
"They're all very good players and on a very good batting wicket we don't look at that too much and try to focus on the areas that we want to bowl and hopefully that brings the wickets that we're after."