Wright smashed 76 off 43 balls to guide England to 149-4 in 18.5 overs after New Zealand was restricted to 148-6 in Group 1 match.
Fast bowler Finn never allowed New Zealand to build momentum by claiming 3-16 and the Kiwis owed their total largely to James Franklin's 33-ball 50.
England's top order had crumbled in the previous two games, but Wright and Eoin Morgan (30) batted with plenty of aggression in an 89-run match-winning third-wicket stand in 10 overs.
It was the second successive loss for New Zealand after losing its opening match in a one-over eliminator against Sri Lanka. West Indies defeated England by 15 runs in the teams' first match.
"It was nice to bounce back," said England captain Stuart Broad. "Steven Finn set the tone with the new ball. The wicket was a lot drier and it did turn a bit."
New Zealand struck through spinners Daniel Vettori and Nathan McCullum, who clean bowled both openers — Craig Kieswetter (4) and Alex Hales (22).
But Morgan and Wright ensured there were no more hiccups in the middle order with some clean hitting. Wright hammered five fours and five sixes while Morgan was more subdued in his nearly run-a-ball knock that featured a four and a flat six over mid on.
Morgan was brilliantly caught close to the wide mid on boundary by a diving Doug Bracewell in the 17th over and Wright was caught in the covers by captain Ross Taylor when only seven was needed for victory.
"I thought it was a decent score but not enough at the end of the day," said Taylor. "We had our moments but we couldn't capitalize on them."
Earlier, Finn took the wickets of Martin Guptill (5), Brendon McCullum (11) and Taylor (22) as New Zealand struggled to put on a partnership after winning the toss and opting to bat first.
Guptill was trapped leg before wicket off a full-pitched delivery and Brendon McCullum sliced Finn to third man where Wright took a comfortable chest-high catch.
Taylor put on 40 runs with Franklin off 28 balls and just when he looked for acceleration Finn had the Kiwi captain caught at deep midwicket in the 17th over.
Franklin rebuilt the innings by smashing four boundaries and two sixes in his 50 off 33 balls and added 39 runs off 21 balls with Nathan McCullum before he was run out in the last over. Nathan McCullum blasted Stuart Broad for two sixes in one over to remain unbeaten on 16.
England shuffled its bowling lineup and brought in Danny Briggs and Tim Bresnan in place of Jade Dernbach and Samit Patel. Briggs justified his selection for the first time in the tournament by bowling to a tidy line and length with the new ball and switched ends to get the wicket of Kane Williamson (17).
However, Franklin blotted the leftarm spinner's figures by hitting 16 runs in his last over.
New Zealand was dealt a setback when allrounder Jacob Oram fell ill and was replaced by fast bowler Bracewell.
New Zealand takes on West Indies in its last group game while England meets Sri Lanka on Monday.