New Zealand batted the final day needing a substantial innings to avoid defeat and were helped by sloppy England fielding.
Rain allowed Williamson and Taylor just enough time to reach their personal milestones in a match clearly headed towards a stalemate.
The century is Taylor's 19th in Tests, second only to Williamson among New Zealanders.
That result had looked inevitable regardless as, on a heartbreaking pitch that barely looked used even by the fifth day, New Zealand had moved to 241 for two in their second innings - a lead of 140 runs - and the chances of a collapse looking minimal.
The hosts hold a 110-run lead and barring an almighty collapse and a quick run chase, or a fearless declaration and a batting capitulation, the series will go to New Zealand following their innings win in the first Test.
Denly dropped New Zealand captain Kane Williamson before lunch at Seddon Park from the type of chance you'd be horrified to see go down while watching an under-eight game.
Williamson was earlier dropped by England stand-in wicketkeeper Ollie Pope when on 39 and went to lunch four short of his 21st test century.
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But while Archer was celebrating this apparent breakthrough with a scream, Joe Denly was looking down at the ball on the grass in front of him.
During the post-match press conference, Root was asked about the incident and he said: "It was a one-off goober".
It drew instant comparisons on social media to the widely regarded worst dropped catch in Test history when England's Mike Gatting put down a sitter off the gloves of India batsman Kiran More in 1993. We worked very hard to try and create chances on a very flat wicket and they don't come around very often.
Williamson had another life when he should have been run out on 97 but Sam Curran's shy at the stumps was so far off line that the batsmen picked up an extra run.
As Root kept himself at the bowling crease, Taylor launched the part-time off spinner for four through midwicket, before two trademark sixes saw the 35-year-old move to 99, and then 105.
They will head to Perth on Saturday to play a three-test series against Australia, starting with a day-night pink-ball test at Optus Stadium in Perth, starting on December 12.