Williamson top-scored with 72 from 46 balls and opener Martin Guptill contributed 65 as the Black Caps battled to a tight victory over England.
The win has taken New Zealand to the second place in the tri-series while England languish at the bottom after losing all the three games they have played that include two against Australia.
Guptill and Williamson made up for the loss of Munro with an 82-run partnership, during which Guptill brought up his half-century. Chapman, now a dual global, became the first player born in Hong Kong to play for a full member side since Dermot Reeve's England debut in May 1991.
Chasing 196 to win their first game of the T20 Tri-Series, England were 79-1 after eight overs but lost their way after Dawid Malan (59) and Alex Hales (47) were both dismissed. Dawid Malan fought a lone battle with a valiant 59 and the lower order threatened, but Sodhi, Mitchell Santner and Trent Boult took two wickets each to wrap the game.
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New Zealand can now secure a spot in the tri-series final with a win over Australia at Eden Park on Friday. Australia are already through to the February 21 final.
Jos Buttler was England captain, with Eoin Morgan still sidelined by injury, and he put the Kiwis into bat because of the uncertain nature of the drop-in pitch at the Westpac Stadium in Wellington.
Continuing their stand-out form from the domestic T20 competition, Auckland's Mark Chapman made a confident and composed 20 off 13 balls, while Northern Districts wicketkeeper Tim Seifert blasted an unbeaten 14 off just six deliveries in a critical late cameo to push the Blackcaps total just shy of the 200 mark.
New Zealand took 62 from the last five, Williamson making Wood curse his earlier error even more by taking 17 off his penultimate over.
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The brio has gone out of Rashid, who was fortunate to pick up his two wickets, the first with a full toss, the second with a superlative one-handed leap at long-off by Chris Jordan whose foot, when he eventually landed, was only an inch from the rope.
Vince was run out by Williamson for 10, Buttler hoisted Sodhi straight to long-off and Sam Billings left no impression. We got plenty in the power play and there partnerships throughout. "It was important we assessed the wicket, [it] played a lot better than it looked and we got off to a good start".
"You always want more runs, you always train hard, that's the focus", he said.
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The offensive comes ahead a March vote that will undoubtedly see Mr El Sisi win a second four-year term. They are blamed for a series of bomb, gun and suicide attacks that have killed hundreds in Egypt.