England were holding on by a thread at the close as Prior sought to shield last man in Monty Panesar from the bowling for the last three overs in a cliff-hanger finish to the Test and the series.
At the close, Prior was unbeaten on 110, his seventh Test century, and England were 315 for nine.
A despondent New Zealand were left wondering what the outcome would have been had two regulation catches in the final over before lunch not been dropped.
All three Tests ended in draws, denying New Zealand the chance of a rare series win over England. They have only won three of 32 series against the English and the last was in England in 1999.
New Zealand, with innings of 443 and 241 for six declared, batted England out of the Test and the tourists were never serious about chasing their 481-run second innings target.
Instead they faced a battle for survival and when they resumed the final day at 90 for four the odds were heavily in New Zealand's favour.
At lunch, the Test still belonged to New Zealand with England at 158 for five and Joe Root gone for 29.
But there were signs of luck going England's way with Ian Bell and Jonny Bairstow both surviving catch opportunities when New Zealand took the second new ball just before lunch.
Bairstow was gone for six soon after play resumed but Bell made the most of his opportunity as he and Prior put on 78 before he was removed in the last over before tea and England were 237 for seven.
But luck stayed with Prior who beat two appeals for caught behind that went to review on 18. On 20 he was dropped by Neil Wagner and on 28 the ball deflected off his helmet and on to the stumps but the bails did not come off.
As he continued to march towards his seventh Test century after tea he was joined by a record low-scoring Stuart Broad who blocked and shouldered for 61 deliveries and 102 minutes before finally getting off the mark.
His scoreless reign beat the previous record of 101 minutes without scoring set by New Zealand fast bowler Geoff Allott against South Africa 14 years ago.
With a possible victory slipping away, New Zealand captain Brendon McCullum again turned to part-time spinner Kane Williamson who had removed Alastair Cook and nightwatchman Steven Finn the previous day.
Again Williamson delivered.
In the space of three deliveries he had Broad and James Anderson caught at first slip by Ross Taylor and England found themselves nine down with only Panesar left to support Prior and 19 balls remaining.
Prior was able to protect Panesar from all but five deliveries which the England spinner was able to keep out.
Mumbai Roundup: 9 incidents that took place this week
Photos: Ajay Devgn, Parineeti, Tabu at 'Golmaal Again' trailer launch
Photos: Dipika Pallikal Karthik oozes oomph in bikinis and sarees
Photos: Kareena Kapoor Khan's grand birthday bash
Photos: 'Dangal' girls Sanya Malhotra, Fatima Sana Shaikh's dinner outing