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England, who finished second in Group 1 after winning three of their four matches, face the unbeaten Kiwis in Delhi on Wednesday.
England's women face Australia earlier the same day for a place in the final.
"New Zealand have played out of their skin in the group stages," England captain Morgan told BBC Sport.
"They have probably played the best cricket of the tournament so far."
Kane Williamson's side qualified from Group 2 in top spot, courtesy of wins over hosts India, Australia, Pakistan and Bangladesh.
England lost their first game, against West Indies, but recovered superbly to win their next three against South Africa, Afghanistan and Sri Lanka.
"There is a a lot of excitement around the group," added Morgan. "Having done enough to get out of the group stages and play in a World Twenty20 semi-final, the guys are really excited.
"We have built a lot of confidence. We also showed a lot of character at different stages.
"We showed our development as a side in chasing down 230, our grit against Afghanistan and we played a lot of smart cricket against Sri Lanka."
England's limited-overs performance in India is a marked improvement on the one they produced in last year's 50-over World Cup, from which they eliminated in the group stages having won just two of their six games.
New Zealand finished runners-up to Australia in that tournament, having played an aggressive, attacking form of cricket.
It is an approach England have attempted to emulate since Trevor Bayliss replaced Peter Moores as coach, with some success, including a 3-2 ODI series win over the Kiwis last summer.
Morgan added: "I think that New Zealand series was very important. We'd talked about emulating what Australia and New Zealand did at that World Cup.
"From where we were to where they were, we were miles away and in order to bridge the gap we had to try and emulate the fashion in which they played and the aggressive nature in which they went about their game.
"So they did play a key part, absolutely."
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England sealed their place in the last four of the World T20 with a 10-run win over Sri Lanka on Saturday, in which Chris Jordan and Ben Stokes impressed with their bowling in the final overs of the reigning champions' chase.
"You want guys to come in and say they want to bowl at the death," added Morgan.
"Ben Stokes and Chris Jordan are guys who I have relied on in the past and they have grown into death bowlers by virtue of their performance.
"It is an area we have struggled with in the past and he [Jordan] has taken on the responsibility of being the man who wants to bowl at the end.
"He always has a cool head, knows his plans and goes about executing them well."
The men's semi-final against New Zealand is the second part of an England World Twenty20 semi-final double-header in Delhi on Wednesday with Charlotte Edwards' women's side taking on Australia in the morning.
"It is a proud day," added Morgan. "The girls have been outstandingly great in the last five or six years.
"They have held the flag very high, even when we have been poor. It is nice to be able to share tomorrow with them. It would be remarkable to both go through to the final."
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England's women won the inaugural Women's World T20 in 2010 - their only triumph in the event.
They have been involved in some extremely close games in this tournament but remain unbeaten, winning all four of their matches to qualify as Group B winners.
"We have learned a lot from our games," England captain Charlotte Edwards told BBC Sport. "We know we can get over the line and it has probably brought us closer together as a team."
Wednesday's game gives England the chance for revenge after being beaten by their old rivals in the 2012 and 2014 finals.
Edwards is one of a number of England players who featured in the most recent Big Bash T20 competition in Australia, but she says any friendships will be put on hold during the semi-final.
"I have played with a few of the Australia girls but as soon as we step over the line... it is the fiercest rivalry in cricket and certainly in the women's game," she added.
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"With the ball England have been ruthless," says BBC cricket commentator Charles Dagnall. "Anya Shrubsole the pick, backed up well by the spinners, but Jenny Gunn has been a standout at the death.
"To progress, it's all about their batting. Head coach Mark Robinson has tried to change the dynamism of the England line-up, and up front it's worked.
"Charlotte Edwards' role quite simply is a boundary hitter. She isn't quick enough to pick up the ones and twos, yet she is the best in the world at piercing the infield during the powerplay.
"Tammy Beaumont has come of age and rather than play second fiddle to Edwards, she has been given the freedom to attack.
"England's middle order struggles on the slow pitches to get going against spin. Sarah Taylor, while her keeping is exemplary, 28 runs in four innings is under par for a player of her class.
"Lydia Greenway's place is under threat but her being the only left-hander in the side will be taken into consideration.
"Spin has dictated the women's WT20, it shouldn't change in Delhi. The winner of this match will be who copes best as both sides prefer the ball coming on. England shouldn't be afraid and I would play them all.
"Here is my team for the semi: Edwards, Beaumont, Taylor, Knight, Sciver, Wyatt, Gunn, Brunt, Shrubsole, Marsh, Grundy."
Eoin Morgan says England will have to beat the in-form side of the tournament in New Zealand if they are to reach the final of the World Twenty20.