The hosts chased down the target in 49.3 overs.
But captain Kane Williamson and Taylor stabilised the ship and, after Williamson gloved a legside catch behind from the first ball that Ben Stokes bowled, wicketkeeper Tom Latham joined Taylor.
That may be optimistic but how New Zealand will want their number four fit for the series decider in Christchurch on Saturday (March 10).
It was just like old times for Bairstow and Root, who have known each other since their early teens.
Taylor's century was his 19th in one day internationals and his highest in 203 matches.
Alongside Williamson, Taylor steadied the innings.
Taylor struck a career-high unbeaten 181 at University Oval, building into his innings before shaking off a thigh issue to thump boundary after boundary.
3- Number of scores in an ODI chase higher than Ross Taylor's unbeaten 181. Unable to run, Taylor freed his arms.
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Morgan was the first man to show any mastery of the pitch, reaching his 48 from 71 balls with three fours and a six - one of only two sixes in the innings.
When Latham fell, miscuing Tom Curran to backward point, de Grandhomme picked up the mantle.
Victory still appeared improbable for New Zealand as they were left needing more than nine runs per over off the final 12.
It was not that simple. In England, one can view matches on Sky Sports Cricket.
Nicholls won the match with a six over square leg. Two towering straight sixes and a late cut were the highlight as he and Root took advantage of an off-colour bowling attack.
When Bairstow, who had been dropped on 74 by Tim Southee, and Root were in full flow an England score of 400 plus and a record total on this ground was on the cards. He may have reached three figures quicker had he not struggled for the strike.
59- Runs scored by England in the last ten overs for the loss of five wickets.
He continued to strike the ball beautifully though and helped bring the rate down to just under eight going into the last 10 overs. "We had to take the game even deeper and it was just a shame not to get across the line tonight". He was superb. But England lost this game.
Ex-England batsman James Taylor: "It looked like complacency - nearly arrogance - from England, to go hell for leather straight up in your innings". "After a tough start, he got himself in and played his best one-day knock", Williamson said. He has been batting like this all year. He had a few niggles. An outstanding performance from him.