In the last game before Popovic picks his squad for the World Cup on June 1, Johan Vasquez's header in the 28th minute proved the difference in Los Angeles.
Australia were dominated by the world No.15 side in the opening half in front of a raucous, pro-Mexican crowd of 78,479 fans at the Rose Bowl, though Mohamed Toure should have equalised on the stroke of halftime.
The Socceroos responded with a vastly improved, threatening performance after halftime, with Ajdin Hrustic having a great chance blocked in the 67th minute.
Popovic lamented Australia's inability to maintain possession before the drinks break on the half-hour mark, while defender Alessandro Circati said the Socceroos were "braver" and more "carefree" in the second half.
"Then after the break we finished the half well, and then the second half, I thought we were very good, which will give us a lot of confidence that we can play a very good nation," Popovic said.
"The players just need to believe a little bit more.
"In the first half we were very anxious and nervous, then after the (drinks break) I thought we saw good signs, and then second half was good. We had the two best chances of the game.
"You're not going to get many chances in football. It's the World Cup you're preparing for - you don't get many chances."
The Socceroos will play Turkey, co-hosts the United States and Paraguay in the group stage.
But first, Australia head to their Bay Area base on Sunday (Monday AEST) before the squad is announced, then will want to be on the front foot in their final friendly against Switzerland in San Diego on June 6.
"Jacko (Jackson Irvine) said it as soon as we came in the dressing room - that's what we've got to be from the first whistle," defender Harry Souttar said of the second half.
Popovic named an experienced line-up, with Mathew Leckie and Souttar making their first starts since late 2024, and indicated no players had drastically changed his mind on selection.
"This will be difficult straight after the game, but I wouldn't say that we had several players needing to do something extraordinary today to get picked," he said.
Fresh off defecting from Italy, Cristian Volpato unsurprisingly didn't make the squad, having only arrived in camp from Australia on match morning.
Joe Gauci, Brandon Borrello and Tete Yengi didn't make the 15-man bench, while Jason Geria, Kye Rowles, Milos Degenek, Cameron Devlin and Martin Boyle were unused substitutes.
Leckie started as an inverted right winger, while Souttar was in the starting back three alongside 18-year-old Lucas Herrington and Alessandro Circati.
Herrington, Souttar, Jordy Bos and Toure were among the bright sparks.
In the seventh minute, Toure was incensed when brought down by Edson Alvarez - which Popovic indicated would have been a red card in a World Cup game.
Mexico scored when Alexis Vega whipped in a corner and Vasquez ghosted in behind Aiden O'Neill, jumped and turned a bullet header on to the post to beat Ryan.
In stoppage time, Toure pounced on a poor defensive header from Mateo Chavez Garcia but chipped wide of the open goal mouth.
The stadium was in raptures when legendary goalkeeper Guillermo Ochoa, headed to his fifth World Cup, entered at halftime and three minutes later batted away O'Neill's fierce long-range strike.
Saturday's "friendly" erupted into an all-in scuffle after Mexico attempted to take a free kick quickly and thought they'd scored in the 76th minute - but play was called back.
Popovic heavily rotated as the half went on, but Mexico were able to resist a late Socceroos flurry.
