Villarreal's half center opened the scoring one minute before half-time when he got his head to Paulo Dybala's free kick as the Mexican defender stood up and watched.
Then Brizuela, who replaced Miguel Layun just three minutes earlier, made his own goal eight minutes from time to ensure the home side's victory.
Both teams are led by temporary coaches and Argentina once again without Lionel Messi, who took a sabbatical from international football after the World Cup finals in Russia this summer.
But it was Lionel Scaloni, who took over after Jorge Sampaoli's short job ended with a disappointing World Cup appearance, which was on top to record his third win in five matches and his fourth clean sheet.
Argentina and Mexico meet again in Mendoza on Tuesday.
(Reporting by Andrew Downie; Editing by Ian Ransom)