The Kiwis went into the match as favourites, as the only remaining unbeaten team in the tournament.
England, by contrast, were defeated by West Indies in their opening game before securing wins over South Africa, Afghanistan and Sri Lanka.
Colin Munro (46) gave New Zealand a strong start, powering them to 1-89 at the halfway stage before Eoin Morgan’s side took control with some smart catching and tight bowling.
New Zealand managed only 64 runs in the final 10 overs and 20 from the last four sent down by Chris Jordan and Ben Stokes.
“We got off to a good start but we were 20-25 short,” Williamson told reporters.
“England bowled really well in that death period which made life very difficult.
“Unfortunately we were not able to connect as well as we’d have liked. It certainly was not through a lack of trying.”
For a team packed with big-hitters, New Zealand could not press hard enough on the accelerator pedal as England claimed regular wickets.
“We were 130 for three (in the 16th over) which certainly is a very good platform in any match if you’re playing T20 cricket and we were not able to capitalise on it,” added Williamson.
England’s players are reaping the benefits of being given a licence to thrill by coach Trevor Bayliss, said Jason Roy following the win.
After their bowlers restricted the Kiwis to a below-par 153 for eight, Roy’s belligerent 78 helped the 2010 champions romp home with 17 balls to spare.
England’s new-found aggression under the guidance of former NSW coach Bayliss was evident in the way Eoin Morgan’s side began and ended their run chase at the Feroz Shah Kotla Stadium.
Opening batsman Roy set the tone with four boundaries in the first over from Corey Anderson before Jos Buttler (32 not out) sealed the win in style, hitting a six off spinner Mitchell Santner.
“That’s the way a lot of us play,” man of the match Roy told the presentation ceremony.
“The whole team has got that kind of licence, that kind of opportunity to go out there and do what they want and be free with their skills.
“When you get off to a good start…there’s no reason why you shouldn’t just keep going. I did keep going and got a bit of luck,” added the 25-year-old after compiling a career-best score in a T20 international.
“It was not an idea to try and smash every ball. Sometimes when you go in you struggle and you don’t get a boundary in your first 10 balls and it’s tough for an opener.”
It was a good all-round display by England, who now meet hosts India or West Indies in Sunday’s final, but Roy refused to call it a perfect performance.
“Obviously there are a few bits we need to work on,” he said. “We had to claw our way back after they made a good start, credit to the boys for dragging us back.”
For the second time in three matches, Morgan fell for a first-ball duck but Roy said his skipper was due a big score.
“He is a great captain, he has got a great work ethic,” added the opener. “To work under someone like that gives you a huge amount of confidence.
“If he gets a good score to win us a game, no one would even remember what happened in the previous game. He is… someone a lot of our boys look up to.”
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