The clash, played in front of 78,479 spectators at the Rose Bowl in Pasadena, California, saw Mexico enjoy the lion’s share of possession. Their winning goal came via a Johan Vásquez header from a set-piece.
But Australia arguably created the better chances and should have scored at least once.
The result follows Mexico's 2-0 win over Ghana last week, with coach Javier Aguirre set to announce his final World Cup squad on Sunday.
Mexico have one more friendly against Serbia on Friday (South African time) before they face Bafana Bafana in the World Cup opener in Mexico City on June 11. The game against the Socceroos will make for interesting viewing for South Africa coach Hugo Broos.
Bafana's other Group A opponents, the Czech Republic, are in action on Sunday against Kosovo in their first warm-up match. They have not played since beating Denmark on penalties to secure their World Cup place in March.
They also have another warm-up game lined up against Guatemala in New Jersey on Friday (SA time) before opening their Group A campaign against South Korea on June 12 (SA time).
The Koreans secured a resounding 5-0 victory over Trinidad & Tobago in their friendly clash on Saturday.
Veteran forward Son Heung-min scored twice in the first half before further goals from Cho Gue-sung (two) and Hwang Hee-chan.
South Korea had 19 shots at goal, 11 on target and five big chances, while restricting Trinidad & Tobago to a single effort on target in an utterly dominant display.
They have one more friendly against El Salvador on Thursday (SA time) before their World Cup opener.
Bafana have a final warm-up clash against Jamaica scheduled for Friday, with the venue and kick-off time still to be confirmed.
Jamaica beat India 2-0 before losing 3-0 to Nigeria in friendly internationals this week.
