"I'm very happy ...(though) we have a lot of work to do," Mexico coach Miguel Herrera told reporters through an interpreter. "Going to the World Cup is totally different than playing a game and we have a lot of work to do before that.
"It was very tough for us to make it.
"We have a lot of work to do on and off the field but we will mix both together and I think Mexico will do well in the World Cup."
Peralta had put their World Cup plans beyond doubt with a 14th minute goal before he completed his hat-trick with further strikes in the 29th and 33rd minutes to cement Mexico's sixth successive qualification, and 15th finals appearance.
New Zealand midfielder Chris James slotted an 80th-minute penalty for the home side after Mexico captain Rafael Marquez handled the ball in the area before Rory Fallon grabbed a second three minutes later.
Carlos Pena, who had been a danger the entire game, grabbed his side's fourth with three minutes remaining in regular time.
The qualification ended a tumultuous period in Mexican soccer that many of their fans would sooner forget.
El Tri had lurched through qualifying, winning just two of their 10 games in the final six-team final phase in North and Central America and the Caribbean and turned over coaches with stunning regularity.
The avuncular Herrera was the fourth head coach appointed to the role in the space of six weeks when he took on the job last month and he was appointed just for their playoff clashes with New Zealand.
While Herrera had predicted his side, picked solely from the domestic league, would comfortably account for the All Whites, much more was at stake than reinforcing national pride.
Sports business experts had earlier told Reuters they had projected a loss of $650 million in broadcasting rights, sponsorship agreements, apparel sales and brand value of the game in Mexico if the national team did not qualify for the finals in Brazil.
Organisers are also no doubt pleased the more commercially attractive North Americans had made the tournament with up to 50,000 fans from the country expected in Brazil.
They had been minutes away from missing out on the playoffs altogether when they were beaten by Costa Rica last month but survived only after Panama blew a 2-1 lead against the U.S., who qualified automatically along with Costa Rica and Honduras.
Their attacking touch, however, returned against the All Whites with their pace and deft touch providing numerous scoring opportunities with Pena and Miguel Layun splitting the defence seemingly at will.
"I think we defended poorly and we allowed to get ourselves exposed pace wise," New Zealand coach Ricki Herbert said of Mexico's three first half goals.
"I think the last 45 minutes were quite refreshing."
Herbert, who is stepping down after eight years in charge, had selected a more attacking side for Wednesday's clash following their Azteca mauling.
The home side showed plenty of intent and endeavour but their best first half chance was when striker Jeremy Brockie won a penalty.
Brockie took the spot kick himself but Moises Munoz pulled off a fine diving save to deny the 26-year-old his first international goal.
- Sports & Recreation