All-rounder Wright's 20-ball innings of 42 included three fours and four sixes as England's power-packed batting line-up scored 214 for seven, their highest total in a T20 international on Saturday.
"The freedom the guys played with the bat today set us up brilliantly," Broad said in a televised interview. "To go and set our record as an England batting line-up was fantastic.
"It was exciting to see the cleanness with which the guys were hitting it."
The 27-year-old Wright then put immense pressure on New Zealand's batsmen as he bowled several dot balls that forced the hosts to take risks, one of which resulted in Martin Guptill blasting a full delivery straight to Broad at mid-off for 44.
Body blow
Guptill's wicket in the 13th over was a body blow for New Zealand as the right-hander had looked well set to be the batsmen the hosts could build their late-innings assault around.
Broad then chipped in with two wickets in the 15th over, removing James Franklin (eight) and Colin Munro (28) to ensure England's total was well out of reach.
The England skipper finished with 4-24, his best T20 international figures, while Wright took 2-29 and added a catch as New Zealand made 174-9 off their 20 overs.
"I think we learned a lot from batting on it first actually," Broad added of being asked to bat by New Zealand captain Brendon McCullum.
"We learned that we could bang it back of a length, take the pace off and get the guys hitting to the bigger boundaries.
"It was certainly a good toss to lose."
Poor day
The short boundaries straight down the ground, which is primarily configured for rugby, were peppered with a total of 23 sixes dispatched into the stands.
The highest total for sixes in a Twenty20 international was the 24 hit by New Zealand and India in Christchurch 2009, a feat duplicated by Australia and India in Bridgetown a year later.
Eoin Morgan top-scored for England with 46 from 26 balls and combined with Jonny Bairstow (38) in a 81-run partnership off 7.1 overs as England appeared set for a total in excess of 230 on the bouncy, drop-in pitch until the stand was broken.
New Zealand did not help their cause by dropping five catches.
"It was a really poor day in the field," McCullum said. "You can't afford to drop five catches in the field in any form of cricket, particularly in 20 overs with the power they have got.
"I thought England were outstanding and put us under pressure. We were outplayed by a far better team on the day."
The second contest of the three-match series is in Hamilton on Tuesday.