Rodriguez struck wonderfully after 98 minutes
Mexico took a surprise lead after five minutes when Rafael Marquez got on the end of a Mario Mendez free-kick.
Argentina replied five minutes later when a corner was forced home by a combination of Hernan Crespo's foot and Jared Borgetti's head.
Rodriguez set up a quarter-final with Germany when he chested Juan Sorin's pass and stunningly volleyed home.
The pre-match script had Argentina stroll written all over it.
Jose Pekerman's team were the outstanding side in the tournament's opening stage, while Mexico were far from convincing in progressing from a weak group.
606 VIEW It would have been a tragedy if Argentina had gone out this early
Dave
But Mexico showed they were not in Leipzig to make up the numbers from the kick-off as they threatened to cause the 2006 World Cup's first major upset.
Argentina had already survived a scare before Marquez put them into an early lead.
Mendez's whipped-in free-kick was flicked on by Pavel Pardo and Gabriel Heinze allowed Marquez a free run to the back post, where the accomplished centre-half finished like an expert centre-forward.
The Argentina fans inside the Zentralstadion may have been shocked, but their players kept their cool and within five minutes they were back on level terms.
Borgetti got his head down to bravely attempt to clear a Juan Riquelme corner and was unfortunate to see it deflect past his own keeper.
Crespo wheeled away confidently claiming his third goal of the tournament but it was officially handed to Borgetti, by all except German television.
However, and despite Borgetti seemingly having the decisive touch, it was later credited to the Chelsea forward.
The two-time World Cup winners began to dictate the play with Riquelme pulling the strings and Esteban Cambiasso attempting to unlock Mexico's offside trap.
Borgetti headed into his own net, but Crespo was given the goal
But Mexico showed some resilience of their own and Bolton's Borgetti was a constant handful on the floor and in the air.
The striker, who recovered from a thigh injury to return to the starting line-up, tested Roberto Abbondanzieri with a long-range effort and was poised to pounce when the keeper later dropped a cross.
Argentina received a let-off of huge proportions when Abbondanzieri rolled the ball out and Heinze had a momentary lapse of concentration.
Fonseca leapt at the chance but with a clear run on goal he was felled by Heinze.
Referee Massimo Busacca chose the wrong option by only showing a yellow card to the Manchester United defender.
Mexico's Argentine boss Ricardo La Volpe berated the official from the sidelines and his side were poorly rewarded when Marquez hit the free-kick straight at the wall.
There was more end-to-end action after the break as Borgetti was clean through on goal only to be denied by Abbondanzieri and Javier Saviola was denied by the impressive Sanchez.
The pace of the game inevitably slowed, with Argentina's patient passing game dictating the action.
But even with the introduction of Lionel Messi and Carlos Tevez they found the Mexican defensive wall too sturdy to break down during the 90 minutes, although Messi had an effort controversially ruled out for offside in second half injury-time.
With chances not forthcoming it needed something special and Rodriguez obliged in sensational fashion eight minutes into extra-time.
He brilliantly teed himself up by chesting a Sorin cross-field pass and then unleashed a left-foot volley past Sanchez's despairing dive and into the top corner.
A weary Mexico had no answer but they go home knowing they tested Argentina to the maximum.
Argentina: Abbondanzieri, Ayala, Heinze, Scaloni, Sorin, Cambiasso (Aimar 76), Riquelme, Maxi, Mascherano, Saviola (Messi 84), Crespo (Tevez 75).
Subs Not Used: Franco, Coloccini, Palacio, Milito, Cufre, Cruz, Burdisso, Gonzalez, Ustari.
Booked: Heinze, Sorin.
Goals: Crespo 10, Maxi 98.
Mexico: Sanchez, Salcido, Marquez, Guardado (Pineda 66), Osorio, Castro, Mendez, Pardo (Torrado 38), Borgetti, Morales (Zinha 74), Fonseca.
Subs Not Used: Corona, Suarez, Franco, Ochoa, Bravo, Garcia, Arellano, Rodriguez, Perez.
Booked: Marquez, Castro, Torrado, Fonseca.
Goals: Marquez 5.
Att: 43,000.
Ref: Massimo Busacca (Switzerland).
Fifa man-of-the-match: Maxi Rodriguez
TRIVIA
Rafael Marquez's opening goal marked the 16th match at the 2006 World Cup in which a goal was scored in the first 10 minutes. No other edition of the World Cup finals has contained so many early opening goals.
Hernan Crespo has scored in the last four World Cup matches he has played in. The last game in which he did not score was on 7 June 2002 against England. Jairzinho and Just Fontaine both scored in six consecutive World Cup matches played. Crespo is the first player to score in three different encounters at this tournament. In 2002, Ronaldo scored in six different matches.
Argentina are now unbeaten in their last five World Cup contests which is two matches shy of their all-time record of seven unbeaten on the bounce in 1986 when they won the World Cup in Mexico. This defeat was Mexico's 22nd loss in World Cup history.
Trivia stats source: Infostrada Sports
50px An epic round sixteen contest between Argentina and Mexico was won by a work of genius from Maxi Rodriguez, Saturday.



Juan Sorin hit a cross-field pass to the midfield player at the top right edge of the Mexican box. Rodriguez jumped to receive the ball and, in the same moment, chested high the way it had come. He met the descending ball 25 yards out with a perfectly timed left-footed volley and the ball arrowed over Oswaldo Sanchez into the opposite corner of the net.

The evenness and excitement of the match might easily be overshadowed by the winning goal. Though the shot totals were not high - Argentina had five shots on target to Mexico's three -  the action was end to end.

Part of the reason for the excitement was while Mexico tried to keep a fast attacking pace Argentina attempted to slow the game down. There could have been more goals; both Argentina and Mexico defended well and had timely saves from their goalkeepers. 

Mexico got off to the best possible start with a goal from a set-piece after five minutes. Pavel Pardo's cross from the right touchline was flicked on by Mario Mendez and Rafael Marquez slid in to score on a half volley at the far post.

For Argentina the equalising goal five minutes later was the best possible response to going behind. A corner kick was swung in low to the near post; Hernan Crespo raised his foot, Jared Borgetti stuck out his neck and between them the ball sped past Sanchez.

The match had its moment of controversy. Just before half time the Argentine goalkeeper rolled a ball to Gabriel Heinze. Somehow his defender let it go past him to Mexican striker Jose Fonseca. Heinze lunged to bring Fonseca down. Massimo Busacca adjudged Roberto Ayala to be the last defender and spared Heinze the red card for the deliberate professional foul. 

Goalkeepers at either end made match saving contributions in the second half.  On 53 minutes Roberto Abbondanzieri saved a defensive a clearance that deflected off Borgetti six yards from goal. Five minutes later Javier Saviola was through on goal; he went high and hard with the shot but Sanchez stuck out a strong left palm and batted the ball away for a corner. 

Lionel Messi and Carlos Tevez replaced Crespo and Saviola as Argentina sought to break the deadlock. A minute from time Tevez thought he was through on goal but Ricardo Osorio got back just in time and the game was destined for another 30 minutes and that special moment in extra time.

Argentina's win meant they would play their quarter-final match in the Olympiastadion, Berlin, against the host nation Germany on June 30.

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