Although New Zealand picked up three quick wickets, captain Heather Knight, who returned to the side, and Nat Sciver-Brunt guided England home in the 12th over.
New Zealand skipper Sophie Devine won the toss and elected to bat, which looked a solid call as Suzie Bates and Georgia Plimmer put on 33 through the powerplay.
Leg spinner Glenn, though, struck with her first delivery of the next over to bowl Plimmer and then just two balls later had Amelia Kerr caught at short midwicket by Alice Capsey.
The White Ferns soon lost another batter when Devine was dispatched for just five by an inswinger from Danielle Gibson which clattered through the stumps.
New Zealand were in freefall at 42 for four when Sophie Ecclestone bowled Bates (16) at the end of the ninth over.
Ecclestone had another wicket when Brooke Halliday clipped a length delivery straight to Gibson at mid-on as the New Zealand collapse continued.
Maddy Green and Izzy Gaze finally managed to put some more runs on the board heading towards the final five overs to bring New Zealand on to 80 for five.
It was, though, only a temporary reprieve as Green (24) chipped out to mid-on off Glenn, who then swiftly sent Jess Kerr back to the pavilion for a three-ball duck to collect her fourth wicket.
Gaze struck a six over mid-wicket which flew through the hands of Wyatt on the boundary, but was run out on 25 off the final ball of the New Zealand innings, with Lea Tahuhu unbeaten on nine.
England's openers soon set about the run chase, with Dunkley, who had been dropped by Kerr in the third over, and Wyatt swiftly making a 50 partnership.
New Zealand, though, slowed England's momentum with three quick wickets as first Dunkley (26) top edged a shorter ball from Tahuhu, which was picked out by Bates.
Capsey (nine) was then bowled by Kerr with the final delivery off the seventh over before Wyatt (26) was caught at short fine leg from the first of the next by Eden Carson to put England at 73 for three.
It was left to Sciver-Brunt, who finished 17 not out, and Knight, unbeaten on 14, to bring England home past their victory target at 104 in the 12th over.
England will now look to complete a clean sweep in the final match of the T20 series at Lord's on Wednesday evening.
"I think we played really well. The crowd were great, gave us great energy," Knight said at the post-match presentation ceremony.
"We have worked hard, worked on our skills to repeat consistently. We were ruthless again, which was really pleasing to see. In the next game, we want to continue and finish on a high.
"Competition for places is driving us forward which is a good place to be, and we want to take momentum into the World Cup."