Tevez was offside when he put the twice champions ahead in the 26th minute but there was no argument with his second goal, a 25-metre drive into the top corner early in the second half.
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Gonzalo Higuain, taking his tournament-topping tally to four, also scored after a defensive blunder as Argentina won the second round clash at Soccer City to progress to a meeting with Germany in Cape Town on July 3.
Argentina face the Germans, who had earlier eliminated England 4-1 with the help of another controversial refereeing decision, for the second successive finals having lost to them on penalties in their 2006 quarter-final in Berlin.
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They will be looking for revenge against a team they beat in the 1986 final and lost to four years later when coach Diego Maradona was their captain.
Tevez opened the scoring after his low effort was blocked by goalkeeper Oscar Perez. Lionel Messi chipped the rebound to his fellow forward who had strayed into an offside position and headed the ball into the net.
The Mexicans protested the decision and Italian referee Roberto Rosetti ran over to consult his assistant linesman, who was surrounded by irate players, before confirming the goal.
There was a heated exchange between the teams' two benches over the issue after the half time whistle.
Seven minutes later defender Ricardo Osorio clumsily lost control of the ball as he was bringing it out of defence and Gonzalo Higuain pounced to round Perez and put Argentina two up.
"They are two errors that changed the course of the match dramatically and drastically. We lost our shape," Mexico coach Javier Aguirre told the post-match news conference.
WONDER GOAL
Tevez put the match beyond the Mexicans seven minutes into the second half with a drive from 25 metres into the top corner of Perez's net, a wonder goal to match Maxi Rodriguez's winner in Argentina's 2-1 extra-time victory over the same opponents at the same stage in 2006.
"Diego asked me to play up front more, to drop back to help defend when it was right, but to be more of a striker," Tevez, voted man of the match, said.
Tevez admitted to a moment's irritation at being substituted midway through the second half when Maradona sent on Juan Sebastian Veron to help in a holding role in midfield.
Mexico had been equal to Argentina up to the opening goal, a shot from distance by Carlos Salcido coming back off Sergio Romero's bar before Andres Guardado shot just wide of the far post from the right minutes later.
With Giovani dos Santos doing some penetrative running on both wings, Mexico had plenty of pos
