Finn's figures of 3-16 from four overs were the best by an Englishman in the short history of the competition, but late hitting from Franklin and Nathan McCullum sneaked the Southern Hemisphere side up to a competitive total.
New Zealand, who lost their opening Super Eights match to Sri Lanka, began their defence promisingly with the wickets of Craig Kieswetter (4) and Alex Hales (22) in the powerplay.
However, Wright's 76 from just 43 balls, which included five maximums, inspired a 91-run third-wicket partnership shared with Eoin Morgan, who contributed a run-a-ball 30.
Both would fall before victory was secured but Jos Buttler and Jonny Bairstow saw England home with more than an over to spare.
England will progress to the semi-finals if they beat Sri Lanka and the hosts lose to the West Indies this evening.