Bangladesh and New Zealand head into their final group match of this year's Champions Trophy desperate for victory but knowing even a win in Cardiff on Friday may not take either of them into the semi-finals.
Joe Root top scored for England with a battling 64, while Buttler, a specialist in the one-day format of the game, made a quick-fire unbeaten 61 as the home side made a total of 310 all out.
The desperation for a win aside, both New Zealand and Bangladesh would hope for an England victory against Australia, assuming rain does not disturb their calculations again. They were on track at one stage but the uneven bounce on the wicket dismissed Williamson in the 30th over.
England captain Eoin Morgan: "Our batting performance was probably par or below par - so pretty average".
The result ensures England progress to the last four and means they can afford to lose their final Group A encounter with arch-rivals Australia at Edgbaston in Birmingham on Saturday (June 10).
Chasing an imposing 311, New Zealand did not have the best of starts as they lost Luke Ronchi in the first over, bowled by Jake Ball (2/31). He's playing fantastically. On a personal note, I'm over the moon for him.
Wood and Ball, who also removed Ross Taylor for 39, went for less than four runs an over as Rashid's leg-spin was also economical and accounted for Neil Broom and Mitchell Santner.
"I think if we're truly going to be contenders for this tournament, we need to beat the best teams - and Australia are one of the best teams", he said.
Guptill said his role as an opener was to provide some stability and set up a platform to launch the New Zealand innings. The trio weren't overly explosive with the bat but steadily added to the score, as the Black Caps couldn't find the wickets they desperately needed.
But the Black Caps looked more than competitive when Williamson and Taylor came together and it took an outstanding delivery from Wood to end a potentially match-winning partnership, when a ball that ballooned up at the skipper was taken by Buttler.
"We go into every game looking to win".
"He's a very good bowler, unique in the fact that he bowls at such good pace off maybe a five to 10-metre run-up - which you don't see too much around the world", said Williamson.