Mexico should have taken the lead in the second minute of the friendly as Raul Jimenez hit the bar with a diving header from Marco Fabian's cross. Less than a minute later, Funes Mori's slip allowed Alan Pulido to play Fabian through on goal, but his tame shot was saved by Agustin Marchesin.
After surviving those early scares, Argentina settled into the contest and slowly began to threaten through Paulo Dybala and Lautaro Martinez, although they remained vulnerable, with Leandro Parades' inch-perfect sliding tackle inside the area denying Jimenez on the counterattack.
The pressure continued to grow, Guillermo Ochoa making a smart save to keep out Martinez's downward header from a Dybala free-kick in the 38th minute.
And when the deadlock was eventually broken, it was another Dybala set-piece that did the damage, Funes Mori meeting his delivery from a free-kick unmarked to head home the opener.
Dybala sent a well-weighted chip narrowly over the bar in one of few chances of note during a disjointed second half, substitute Brizuela putting through his own net in the 83rd minute to prevent the lurking Mauro Icardi from opening his senior Argentina account.
What does it mean? Jury out on both coaches
Lionel Scaloni can rest slightly easier, at least for now, after a mixed start to his reign as Argentina coach, having overseen a narrow loss to Brazil, a draw with Colombia and unremarkable wins over Guatemala and Iraq. Mexico continue to search for Juan Carlos Osorio's long-term replacement, interim boss Ricardo Ferretti having now orchestrated just one win over Costa Rica and four losses, to Uruguay, the United States, Chile and now Mexico.
Funes Mori makes amends
A careless early error having gone unpunished, Villarreal defender Funes Mori atoned for his sins just before half-time, shrugging off the ineffectual attentions of the Mexico defence to finish confidently from close range, leaving Ochoa stranded.
Dybala can't find his shooting boots
Dybala was by far Argentina's brightest spark before being withdrawn in the 72nd minute, the Juventus forward supplying numerous chances for team-mates and testing Ochoa himself with an eye-catching combination of chest control and a scissor-kick volley. But the 25-year-old is yet to score for his country, and the pressure will continue to mount on his slender shoulders during the continuing absence of Lionel Messi.
The teams will sign off on 2018 when they meet again in another friendly in Mendoza on Tuesday.