The Argentine defender's goal proved the difference in the first half when he got his head to Paulo Dybala's free kick in the 44th minute, as Mexico struggled to put up any defence.
Then in the 83rd minute, Brizuel, who had only entered the pitch three minutes earlier, found his own net at the Estadio Mario Alberto Kempes to give Mexico few positives from the first of two meetings with La Albiceleste.
The visitors had started brightly as Raul Jimenez almost opened the scoring after two minutes with a header that bounced off the crossbar, before Agustin Federico Marchesin was forced to scramble to deny Marco Fabian's follow-up shot.
But the fearless start prompted a response from the youthful Argentina squad, with Giovani Lo Celso attempting a long-distance strike that fell wide after 11 minutes and Marchesin blocking Dybala in the 26th minute.
The first half looked like it would be a goalless affair as Guillermo Ochoa kept Argentina at bay until Funes Mori found a way through with a fine header just before the break.
Mexico made four changes at half-time, Alan Pulido making way for Angel Zaldivar, who was quick on the attack but saw his header go over the bar 12 minutes into the second half.
Mexico continued to have little answer to Argentina's attacking momentum and with seven minutes before the final whistle, Brizuela found his own net to cap off a deserved win for the home side, who will face Mexico again on Wednesday in Mendoza.