England, who had already qualified for next week's semi-finals, topped Group Two by winning all three of their games. New Zealand must wait for the result between Pakistan and Sri Lanka on Saturday to see if they advance to the last four.
"It was on a bit of a knife edge and I'm over the moon," said player of the match Jacks, who took two wickets before scoring a quickfire 32.
New Zealand chose to bat first and although Jofra Archer bowled a maiden first over, openers Tim Seifert and Finn Allen picked up 54 runs in the powerplay.
The English spinners reined back the 2021 finalists, however, with Adil Rashid getting Seifert (35) stumped and Allen (29) falling to Jacks four balls later.
New Zealand did not hit a boundary between the 13th and 17th overs, as Rashid (2-28) dismissed Mark Chapman (15) and Daryl Mitchell (3) fell to Liam Dawson.
Glenn Phillips made a rapid 39 and Cole McConchie (14) added late runs before captain Mitchell Santner hit a six off the last ball to get New Zealand to 159-7.
JACKS AND REHAN SNATCH LATE VICTORY
New Zealand started well, Matt Henry bowling a perfect delivery to get Phil Salt (2) caught behind, before Lockie Ferguson dismissed Jos Buttler for a duck.
With England reeling at 2-2, captain Harry Brook set about stabilising the innings with Jacob Bethell.
They added 48 runs for the third wicket before Brook, fresh from his brilliant hundred against Pakistan, fell for 26 trying to hit Phillips over long off.
Bethell (21) pulled Ravindra to deep mid-wicket, where Phillips dived forward to take a spectacular low catch and leave England in trouble at 58-4.
Tom Banton and Sam Curran kept England in the game with a solid partnership before the latter holed out.
Rachin Ravindra (3-19) found sharp turn and had Banton (33) caught at mid-wicket, leaving England needing 43 to win from the last three overs.
Rehan (19) and Jacks hit Phillips for a six each to pick up 22 runs from the 18th over and they closed out the victory with ease.
Pakistan need to beat Sri Lanka by 64 runs or chase down a target in around 13.1 overs to deny New Zealand a place in the semi-finals.
