Goals by Manchester City's Tevez and Real Madrid's Gonzalo Higuain, his fourth at the World Cup to become the tournament's top-scorer, means Diego Maradona's Argentina will now meet Germany in Cape Town on Saturday in the last eight.
Mexico forward Javier Hernandez gave his side brief hope when he turned his marker in the 71st minute and smashed his shot past Argentina goalkeeper Sergio Romero, although it was too little, too late.
"This was an important step, we're in the quarter finals and that's what matters, that we played well," he said.
"The first couple of minutes were really difficult for us. We were not able to run the match. That made life hard for us. After that, it only got better.
"We need to rest and enjoy this victory, which wasn't easy against Mexico. As for my goals? Well, I just want to be effective.
"Now we have to relax a little ahead of going up against the Germans in the quarter-finals, four years after losing to them" also in the last eight, when the Germans won on penalties.
