The only other times El Tri have made the final eight of the competition was when the tournament was held in Mexico in 1970 and 1986.
On both of those occasions they failed to make it to the final four.
They face the toughest test to reach that mark again when they face the form side of the competition from the group stages in Argentina.
The South Americans emerged group winners from the "Group of Death" and their 6-0 demolition of Serbia and Montenegro remains the performance of the World Cup thus far.
However despite people claiming that this could be the year to end the two time champions' 20 year drought, Mexican defender Rafael Marquez is confident his side can prove a point to the world and their own fans.
"There is potential in the team, but we must also learn from our mistakes. Now the World Cup is really under way but it really hurt to hear our own fans boo us," the Barcelona player said.
Dynamic duo: Maxi and Messi
Argentina coach Jose Pekerman was also out to show some respect to his opponents.
"I have a very positive image of Mexico," the 56 year old said.
"They are high in the FIFA rankings and played well at the Confederations Cup last year.
"It will be a good match. They have many very good players. They will be a strong opponent, and it's the World Cup, so every game is difficult.
"Mexico are always able to score goals," he added. "They are technically skilled and highly competitive team. No team can ever believe that Mexico is not a strong rival."
Argentina’s starlet Lionel Messi will be looking for a double celebration on Satuday as turns 19 on the same day as he looks to help guide his side into the quarterfinals.
The Barcelona player will be hoping he had done enough to earn a place in the starting side since recovering from a thigh injury at the start of the World Cup.
"I feel fit enough to get through 90 minutes because this is when the real World Cup begins," the teenager who holds Spanish citizenship said.
Pekerman has a selection headache ahead of the match as he looks to find a replacement for Nicolas Burdisso who was ruled out of the match with an ankle injury he suffered during their last match against the Netherlands.
Of more interest to Albiceleste fans is who he selects up front with strikers Hernan Crespo, Javier Saviola, Lionel Messi and Carlos Tevez all in blistering form.
Opposition coach and fellow Argentine Ricardo Lavolpe has declared his side will be ready to score goals after looking sheepish up front in their past two matches.
Ricardo Lavolpe: A nation behind
"On Saturday I'm going to have a different team. They're going to be much more at ease and that will help them to score goals," the former Argentine goalkeeper said.
The goal scoring chances increased dramatically when all time Mexican leading scorer Jared Borgetti was declared fit for the sudden death match.
Both teams have said it and both teams are right.
The World Cup really starts now.