Bates' 12th one-day international century and her fourth across four World Cups allowed the tournament hosts to make 8-265 on Saturday after losing the toss at Hagley Oval in Christchurch.
Hannah Rowe then took 5-55, her first five-wicket bag, as New Zealand held Pakistan to 9-194 in reply.
New Zealand came into the match with the faint hope that a convincing win over Pakistan and extraordinary results in the final matches of the group stage on Sunday between England and Bangladesh and South Africa and India might see them just qualify for the semi-finals.
Though they improved their net run-rate, that no longer seems possible. Bangladesh would have to beat England and India would have to beat South Africa by enormous margins for the Kiwis to retain any chance of qualifying.
"Knowing we'd lost that close match to England (last weekend) it was going to be tough to make the semis," Bates said.
"We probably got rid of our emotions in the first couple of days and realised as a team we had a job to do."
Bates has not had the best of tournaments with only one half-century — an unbeaten 79 against Bangladesh. When she reached 81 she passed her 5,000 runs in one-day internationals and went on to a century from 110 balls.
"I think I'm determined every game I play to score runs and it hasn't gone my way this tournament but every time you get a chance to bat you want to make sure you get runs," Bates said.
Bates had sporadic support from Amelia Kerr (24) and wicketkeeper Maddy Green (23) in what is likely to be her last one-day international.
Brooke Halliday hit 29 from 38 balls late in the innings to boost New Zealand's total.
Rowe conceded 27 runs from her first three overs during the power play but returned to take five wickets for 28 runs in an exceptional second spell. Her victims included Nida Dar, who was Pakistan's leading scorer with 50 from 53 balls.
