The Villareal centre half opened the scoring a minute before half time when he got his head to a Paulo Dybala free kick as Mexican defenders stood back and watched.
Then Brizuela, who had replaced Miguel Layun only three minutes earlier, put through his own net eight minutes from time to seal a deserved win for the home side.
Both teams were led by interim coaches and Argentina were once again without Lionel Messi, who took a sabbatical from international football after the World Cup finals in Russia.
But it was Lionel Scaloni, in charge after Jorge Sampaoli's short stint ended with a disappointing World Cup, who came out on top to record his third win in five games and his fourth clean sheet.
Scaloni handed debuts to Spurs defender Juan Foyth and Velez Sarsfield midfielder Gaston Gimenez and was delighted with the performances of what was a young and experimental side that included the return of Erik Lamela after injury.
"I am very happy because some players were able to make their debuts today," Scaloni said. "They are adding something and important things come along with them. The young lads have to know that the jersey carries a weight."
Argentina and Mexico meet again in Mendoza on Tuesday.
(Reporting by Andrew Downie; Editing by Ian Ransom)
