The twice former champions were held 0-0 by Netherlands in their final Group C match on Wednesday and will be looking to find the net again with the ease of their 6-0 rout of Serbia & Montenegro a week ago.
Argentina, with an 8-1 goal record in three group games and two players on two goals apiece in striker Hernan Crespo and winger Maxi Rodriguez, are favourites to reach the last eight.
Mexico have only reached the quarter-finals in the two World Cups they hosted and came to Germany expecting to improve their record abroad.
But their finishing has been so poor they only scraped into the knockout phase.
"We will continue to insist on working on our finishing (in practice)," Mexico's Argentine coach Ricardo La Volpe said after they lost their last Group D match 2-1 to Portugal.
Captain Rafael Marquez said: "The team continue to show poise and character...we have a new opportunity now and we can't let it slip."
The two Latin American nations have only met once before in the World Cup when Argentina won 6-3 in the inaugural tournament in Uruguay in 1930.
They have, however, played each other in numerous friendlies and in the Copa America, to which Mexico have been invited regularly since 1993.
Argentina beat Mexico in the 1993 Copa America final in Ecuador but lost 1-0 to them in the group stage in the last tournament in Peru in 2004.
"Mexico will not be at all easy as opponents," defender Roberto Ayala said. "They have made matters difficult for us the previous times they have played Argentina.
"Mexico have the same chances as us. We'll try to prevent them from being comfortable on the ball and when we attack attempt the variety of options we have showed up to now."
Argentina will have to bring in Fabricio Coloccini after right back Nicolas Burdisso was injured on Wednesday.
Jose Francisco Fonseca, who scored against Portugal, is expected to start again for Mexico even though Jared Borgetti could be fit after missing two matches with a thigh injury.
Brazilian-born Antonio Naelson "Zinha" should replace the suspended Luis Perez, who was sent off on Wednesday.
1-Roberto Abbondanzieri; 4-Fabricio Coloccini, 2-Roberto Ayala, 6-Gabriel Heinze, 3-Juan Pablo Sorin; 18-Maxi Rodriguez, 8-Javier Mascherano, 5-Esteban Cambiasso, 10-Juan Roman Riquelme; 7-Javier Saviola, 9-Hernan Crespo
Coach: Jose Pekerman
1-Oswaldo Sanchez; 16-Mario Mendez, 4-Rafael Marquez, 5-Ricardo Osorio, 3-Carlos Salcido, 14-Gonzalo Pineda; 6-Gerardo Torrado, 7-Antonio Naelson "Zinha", 8-Pavel Pardo; 9-Jared Borgetti, 17-Jose Francisco Fonseca
Coach: Ricardo La Volpe
Referee: Massimo Busacca (Switzerland)