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'I am responsible for this disappointment. It is a great sadness and I assume all the responsibility of this huge failure,' said Martino, whose contract was reportedly due to expire after Mexico's World Cup campaign.
Australia, ranked 35 places below Argentina, have already surpassed expectations in winning twice to earn their place at Qatar's Ahmad bin Ali Stadium this weekend for one of the biggest games in their history.
Poland too progressed, with a better goal difference, after Mexico best Saudi Arabia 2-1.
Mexico finished with four points, level with Poland who lost 2-0 to group winners Argentina and had a superior goal difference of one to advance.
Stephanie Frappart, Neuza Back and Karen Diaz will become the first all-female refereeing team for a men's World Cup match.
The Argentina team did not immediately respond to a request for comment about Alvarez's statement.
After two rounds, Mexico are bottom of Group C with just one point.
Former star Pablo Aimar was Lionel Messi's childhood idol.
The dressing-room party scenes - Messi bouncing with both arms in the air, goalkeeper Emiliano Martinez dancing on a table - showed just how much this tournament means to Argentina and their captain.
The win took Argentina up to second place on three points, one behind leaders Poland, who they play in their last group match.