The 24-year-old's England place has been in serious doubt at times in the last few years due to shoulder injuries and a lack of form.
But, batting in a position he had never previously featured in for England or Surrey prior to the first Test, Pope has repaid the faith of Stokes and coach Brendon McCullum.
Freed of the oppressive shackles of the England captaincy after handing the role to Stokes earlier this year, Root was in vibrant mood himself as he unfurled his full array of strokes in his 27th Test century.
Needing only 116 balls to reach his fastest Test ton, Root has picked up where he left off at Lord's with a second successive century.
His match-winning 115 not out led England to a thrilling five-wicket victory over the Test world champions in the opening game of the three-match series.
Earlier, Alex Lees compiled his highest Test score of 67 before becoming the only England batsman to perish in Sunday's first two sessions.
Daryl Mitchell's Test-best 190 and Tom Blundell's 106 on Saturday had put New Zealand in pole position.
But England were able to replicate New Zealand's success in conditions perfectly suited to batting.
Lees had resumed on 34, while Pope had made 51 not out overnight, with both having survived a dropped catch by Mitchell on Saturday.
Lees, who had a top score of 31 from his first eight Test innings, reached an assured 50 in 109 balls with a composed clip to the boundary.
But, losing his concentration after the drinks break, Lees needlessly chased a wide ball from Matt Henry and edged it to Mitchell in the slips.
Just before lunch, Root escaped when he top-edged Trent Boult to Tim Southee, who allowed the ball to squirm through his hands as he raised them high above his head in a failed bid to make a tough catch.
Within touching distance of his ton, Pope whipped Boult through mid-wicket for four and reached his milestone with a comfortable drive for two.
Joy and relief were etched on Pope's face as he soaked up the crowd's applause and received a warm embrace from Root after his 14th first-class hundred.
Root joined the party with a four to bring up a rapid-fire 50 in just 56 balls.
England were scoring at a blistering pace and Pope's towering six off Henry was the signature stroke of his innings.
The boundaries kept flowing from both players as the stand accelerated past 150.
Root enjoyed a stroke of luck to reach his century with a miscued cut that flashed past his stumps before he waved his bat triumphantly towards his team-mates in the dressing room.