Yet if they continue to underperform, as they did in the CONCACAF group campaign, they risk becoming only the fourth Mexico team in history to fail to qualify for the finals.
Miguel Herrera, the stocky, tough-talking coach, is entrusted with salavaging the nation's pride in the two-leg playoff with the All Whites. The first leg is on Wednesday (2030 GMT) at the Azteca, where Mexico's recent form has been poor, with one win and three goals in four matches.
"We keep telling (the players) what we're playing for, the determination we need, the will to bring back the ticket so Mexico can go to the World Cup," Herrera told a news conference at the weekend.
"On Wednesday we'll be going all-out to get a good advantage. It's important not to concede a goal but also to get a lead," added Herrera, brought in on the basis of a fine year as coach of top Mexican club side America.
"We don't think (New Zealand) will be easy, but they also don't see us in the same light as (teams) they faced in their qualifiers.
The United States have challenged Mexico's status as top dog in the Concacaf region regularly over the past two decades, but Mexico, feeding off one of Latin America's strongest domestic league, have largely taken for granted the task of qualifying for the World Cup.
Only in 1934, 1974 and 1982 have they failed previously, adding to their absence in 1990 because of a FIFA-imposed ban.
This time they reeled from one shock to another and shed three coaches as they finished below the U.S., Costa Rica and Honduras in the six-nation final CONCACAF group and missed out on a direct passage to Brazil next year.
Most damaging has been the Olympic champions' failure to translate success at youth levels to the senior team.
Herrera - who overlooked Mexico's leading European-based players and named his squad from players in Mexico's domestic club football - has two main selection issues.
He must choose a strike partner for Oribe Peralta, who scored both goals when Mexico beat Brazil in the gold medal match at the London Games last year, and a goalkeeper.
Peralta will be partnered by either Aldo de Nigris or Raul Jimenez, while America's Moises Munoz is competing with Olympic custodian Jesus Corona for the number one jersey.
The All Whites' build-up has been less than ideal, with New Zealand Football (NZF) scrambling around for some last-minute preparations against a club side in Los Angeles and a match against Trinidad and Tobago.
"Losing your captain a week before a match of this magnitude is difficult but this is the nature of football," Herbert, whose side was unbeaten in three matches at the 2010 World Cup, said.
"Every player in our squad is relishing the challenge lying ahead and, with or without Winston, the end goal remains the same."
NZF also faced a near-mutiny from A-League side the Wellington Phoenix over the release of players for the squad.
Glen Moss, Leo Bertos, Andrew Durante, Ben Sigmund and Jeremy Brockie all appeared for the Phoenix against Perth - who included All Whites striker Shane Smeltz - in Christchurch on Sunday before all made a dash to the airport to catch a connecting flight to Auckland for their flight to Los Angeles.
All six would be expected to start against Mexico, though the top-scoring Smeltz has only recently returned from hip surgery and played only 53 minutes on Sunday before limping off. Smeltz, however, has said he should be ready to play.
Herbert, a former defender, has been accused of being too cautious and favours a 3-4-3 format with 37-year-old Ivan Vicelich occupying a defensive midfield role in front of three centre backs - Durante, Sigmund and Ipswich Town's Tommy Smith.
That approach might be sufficient at the Azteca, as New Zealand will seek to ensure they are not out of the tie by the time the teams meet for the second leg in Wellington on Nov. 20.