Mexico came into the Alamodome riding a 11-game winning streak under manager Gerardo "Tata" Martino, but his former side emphatically halted that with an offensive outburst from which El Tri ultimately couldn't recover.
Martinez, who plays for Inter Milan, repeatedly took advantage of sloppy Mexican defending during the first half. He slotted in his first in the 17th minute with a slick left-footed shot from just inside the box. He struck again five minutes after getting behind Mexico's line, and added his third in the 39th minute after robbing the ball from Nestor Araujo.
Paris Saint-Germain's Leandro Paredes scored Argentina's other goal from the penalty spot in the 33rd minute.
Mexico fielded a strong starting XI that featured Wolves striker Raul Jimenez and Napoli's Hirving Lozano, but finished the night without registering a shot on goal.
Eight players were shown yellow cards in a match that rarely lived up to its friendly billing, but the game will be remembered largely for the brilliant finishing that brought the 22-year-old Martinez his first international hat trick.
"I'm delighted and emotional," he said. "One has to make a lot of sacrifices to get here and it's not every day that you score three goals wearing this shirt."
The result was Mexico's heaviest defeat since they lost 7-0 to Chile in 2016.
Tuesday's loss came after Mexico had dominated the United States last Friday in a 3-0 win.
Argentina, under Lionel Scaloni, finish the international break on a high note following a scoreless draw against continental rivals Chile last week.