Tuesday's success means England will return to the Welsh capital as Group A winners for a last four clash against the as yet unknown Group B runners-up on June 14, in an event featuring the world's leading eight one-day global (ODI) teams.
The England bowlers then produced a stellar show to dismiss New Zealand for 223 runs by the 45th over.
Asked to bat first, England rode on half-centuries from Alex Hales, Joe Root and Jos Butler to post a formidable total of 310 runs.
England will go through to the semis as Group A winners regardless of the result against Australia, but Saturday's clash could be the final time the two sides meet before this winter's Ashes series.
The Blackcaps have crashed to a heavy defeat to hosts England in their penultimate round robin match at the Champions Trophy.
Morgan feared his team didn't have enough runs to play with after being sent into bat.
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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.
"We all know the importance of getting wickets early in the innings", paceman Boult said at New Zealand's pre-match press conference in Cardiff.
He agrees their tournament is going very well so far, but added: 'If you had asked me that at the halfway stage, I probably would have said it wasn't. "It's a must-win game and as a bowling unit we definitely need to take wickets". Corey Anderson (3-56) and Adam Milne (3-79) shared six wickets between them but Tim Southee was the pick of the bowlers, taking 2-44 from 9.3 overs. Despite a ravishing 48-run knock from Ben Stokes, Eoin Morgan's men were falling short of the 300-run mark.
Meanwhile, all-rounder Chris Woakes is a doubt for the first two Tests of the summer against South Africa after tearing his strain in England's first Champions Trophy game.
"We'll obviously have a chat about the team, but, certainly within a winning side, it's very hard to make changes going into a game where you don't really have much to lose".
"We know that, and I think other teams know that as well".
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