All-rounder Tim Bresnan followed a strong bowling display with an unbeaten 23 off 11 balls as England reached a victory target of 153-7 with five balls to spare.
Earlier, Bresnan's 1-20 had helped limit New Zealand to 149-6 off 20 overs. Ross Taylor top scored for the Kiwis with 44 off 33 balls.
England ended top of the group with six points while Pakistan, New Zealand and South Africa all ended on two points.
Pakistan, the defending champions, advanced to the semifinals along with England due to a better net run rate.
England's chase was given energy by opener Michael Lumb, who struck four fours and a six in 32 off 21 balls.
Lumb was eventually leg before to Daniel Vettori.
Medium pacer Scott Styris claimed two middle order wickets to stall England at 66-4 off nine overs.
Ravi Bopara lofted to extra cover while captain Paul Collingwood clipped a catch to midwicket.
But Eoin Morgan shared a match-altering stand of 52 off 38 balls with Luke Wright.
Left-hander Morgan struck four fours and a six in 40 off 34 balls and Wright compiled 24 off 17 deliveries.
Pace ace Shane Bond removed both towards the end but Bresnan hit three fours to ensure England got home comfortably.
Earlier, Taylor's fourth wicket partnership of 62 with Scott Styris (31 off 19 deliveries) anchored the Kiwis.
Brendon McCullum struck 33 off 32 balls at the top of the order.
Off-spinner Graeme Swann led England's attack with 2-31 off four overs while Stuart Broad took 2-33.
Bresnan removed Jesse Ryder early after New Zealand won the toss and batted.
The left-hander lost his off stump as he swung across the line of a slower ball.
Swann claimed Aaron Redmond and McCullum to stall New Zealand at 65-3 in the 11th over.
Redmond swung to deep square leg where Broad held a brilliant running catch while McCullum, who hit four fours, picked out Michael Lumb at deep midwicket.
Taylor and Styris powered the late innings for the New Zealanders but their total proved insufficient.
