Morgan top scored for the tourists with 48 off 71 while Stokes, the England hero when they won the second ODI three days ago, contributed 39 off 73.
However, a typically-adroit Kane Williamson (112*) century took the match to the final over, yet the New Zealand captain could only muster ten of the 15 runs required for victory, meaning his opponents now move to the penultimate match with a chance of sealing the series.
When time was up New Zealand were 230 for eight in reply to England's 234 after Chris Woakes bowled two dot balls to Williamson to end the match.
"We were in a position of strength at one stage but lost a lot of wickets that hurt us and we had to take the game even deeper", Williamson said.
England lead five-match series by 2-1. It is located in New Zealand.
In variable conditions Eoin Morgan's men battled hard to set 235 as the target for victory, and when their hosts fell to 103-6 in the 25th over they must have been confident of success.
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Their problem was no one could settle in long-term and New Zealand part-time medium pacer Colin de Grandhomme was allowed to bowl 10 overs in which he took one for 24.
A 96-run partnership between Williamson and Mitchell Santner brought New Zealand back into the match, but the skipper rued their blip.
When the teams met in a T20 on the same ground last month, that was the case, the pitch containing more runs, and producing a far better contest, than looked likely.
New Zealand's middle order then failed to deal with England's spin duo - Ali having Mark Chapman (8) caught at backward point, pinning Tom Latham (0) lbw for a golden duck on review and then having De Grandhomme (3) pouched at long-on. The England innings never really kicked off in the game. It took a huge amount of commitment and over-thinking from me and Ben to [work out what a good score is].
It was clear from the moment the first ball Jonny Bairstow received from Trent Boult leapt up at him, taking a piece of the pitch out on its way, and lobbed up off his bat to safety that this was not going to be a run-laden surface. Another wicket followed and England's hold got stronger. It was a very tight game, could've gone either way.
Watch the fourth ODI between New Zealand and England, at University Oval in Dunedin, live from 9.30pm on Tuesday on Sky Sports Cricket (channel 404) and Sky Sports Mix (channel 121).