Root arrived having hit an unbeaten 115 to guide England to a five-wicket win in the first Test but endured nervy moments early on and was lucky to survive after he flashed at a ball from Tim Southee who also dropped him at slip.
But the 31-year-old regained his composure to bring up his half-century off 56 deliveries, before frustrating New Zealand who struggled to contain the batters as they took their team past the 300-run mark in the 75th over.
A reverse sweep off Michael Bracewell took Root to 97 before an edge for a four off Daryl Mitchell two overs later completed his 27th Test century and fourth at this venue, with the knock coming off only 116 balls.
It was Root's fourth century in five Tests, coming after he reached 10,000 runs in the format in the last match, and a flick off the pads for four took him to 150 as he put New Zealand to the sword.
Earlier, it did not take long after lunch for Pope to score his century -- his first since 2020 and first on home soil -- as he cut Matt Henry for four before pushing him through cover for two to draw loud cheers.
Pope fell early in the final session as he top-edged one to Matt Henry off Trent Boult, who then removed Jonny Bairstow for eight after a review showed he edged the ball to the keeper, leaving England at 344-4.
Skipper Ben Stokes' blitz had New Zealand in a spot of bother but he perished for 46 attempting one too many sixes, as Bracewell claimed an important first wicket on his debut.
However, the Black Caps squandered another opportunity when Will Young dropped Foakes on 10.
"We've got to focus on our batting (tomorrow)," Pope added. "Hopefully sneak in front of them or get as close as we can and hopefully put them under pressure with early wickets."
Lees looked in good form earlier, fluently driving Southee for two boundaries before bringing up his maiden Test fifty with a four off Kyle Jamieson, who later sustained a back injury.
On 67, however, Lees chased a wide delivery from Henry to edge it to Mitchell at slip and the fielder made no mistake to end the second-wicket partnership of 141.
The final Test starts at Headingley on 23 June.