With scores locked 1-1 at fulltime the teams had to call on extra energy reserves to battle for a place in the next round.
Eight minutes in to the first period of extra time, a long ball from captain Juan Pablo Sorin found Rodriguez on the right hand corner of the penalty box.
In one motion he chested the ball down to control and then struck an unstoppable volley into the far corner of Mexican keeper Oswaldo Sanchez’s goal to start a celebration of joy mixed with disbelief.
There have been some memorable goals scored by men in Argentinean jerseys at World Cups and this one will be quickly compared to them.
It deserves it be.
In the first half, the Mexicans got their first when captain Rafael Marquez struck after just five minutes.
Marquez opens the scoring
The Barcelona defender found himself unmarked at the far post after Jose Antonio Castro flicked on a Pavel Pardo free kick.
It was just reward for their energetic opening, but they had their advantage taken off them five minutes later.
An Argentinean corner was whipped in by playmaker Juan Roman Riquelme and as strikers Jared Borgetti and Hernan Crespo tangled the ball flew into the CONCACAF runners up goal.
The break neck speed continued as the sides threw everything at their opponents.
Chelsea striker Crespo nearly had a goal that would require no debate about its owner as he broke an offside trap and chipped Sanchez who was advancing rapidly, but it drifted harmlessly wide.
Minutes later Mexico’s record goal scorer Borgetti forced a tremendous one hand save for Albiceleste goal keeper Roberto Abbondanzieri with a rasping drive from the top of the box.
The second half become more of an arm wrestle but players in both sides had their chances to become the hero.
Target man Borgetti came close just nine minutes after the break with a trademark header but Abbondanzieri was up to the challenge.
On the hour dimunitive striker Javier Saviola caught the defence napping to be one on one with keeper Sanchez but the Chivas Guadalajara man stood firm.
Germany here we come
The two time champions almost had a winner in regulation time when replacement striker Pablo Aimar snuck through and slid the ball to Barcelona’s Lionel Messi who duly put the ball in the back of the net.
The referees assistant’s flag halted any celebration, although replays suggested he may have been too eager to play the roll of party pooper.
Justice prevailed though, thanks to a moment of sheer footballing brilliance from Rodriguez.
Argentina will play Germany in Berlin on Friday.