Even then, Williamson nearly stole it in the final over which started with 15 needed and a six over square leg off Woakes made it 7 off 3 - after the match the New Zealand captain said he felt it was within his grasp - before Woakes went full and straight to close it out.
The five-match ODI series is tied at one apiece.
The ugly Westpac Stadium, known locally as the Cake Tin because of its bland, round design, had provided an equally unattractive drop-in pitch that saw England severely tested in battling to a just about par 234 all out.
Mitchell Santner initially steadied the ship before then combining well with his leader, however when he was run-out backing up off the fingers of Woakes for 41, the required 36 from 28 deliveries - with only three wickets left in the pavillion - looked a tough ask.
Rashid also trapped Henry Nicholls lbw with a leg break to put England firmly in control and ensure New Zealand lost 5-23 but Williamson and Santner kept the hosts in with a chance, Williamson sealing his 11th ODI century, from 133 balls, in the penultimate over. We felt in a bit of form [after the second ODI] but it Ben and myself a while to find the middle of the bat.
"We need to play much better against a strong England side".
Whereas victory in the second ODI in Tauranga was an all-round demolition of the Black Caps, this was a streetwise and intelligent adaptation in tricky conditions, even if they were nearly denied by the brilliant Williamson.
Williamson finished unbeaten on 112 but it was England who were celebrating, after a few deep breaths.
Ben Stokes did not hit his first boundary until his 53rd delivery but England knew the pitch meant New Zealand's run chase would be hard.
Earlier, the Black Caps chose to bowl and the seam and spin of de Grandhomme, Munro and Ish Sodhi were effective.
Yet, frustratingly for England, after giving himself a platform for a late assault, Stokes holed out to the longest boundary of this small playing surface when he opened up his arms virtually for the first time. With it the momentum changed. There was fine death bowling from Woakes and Tom Curran for England. Then it was down to the cool head of Woakes to complete a quite tremendous victory for England.