Martinez scored a stunning first-half hat-trick at the Alamodome as Lionel Scaloni's men ran riot in the opening 45 minutes.
Inter forward Martinez was the star, netting three times and also helping Argentina – without the suspended Lionel Messi – win a penalty that was converted by Leandro Paredes.
Martinez came off at the break after putting Argentina in complete control as they saw out a comprehensive victory.
It was quite the way for Mexico to be handed their first defeat under head coach Gerardo Martino, who took over El Tri in January.
Mexico could have been awarded an early penalty for a handball, but instead fell behind in the 17th minute.
Martinez received a pass from Paredes 30 yards from goal, running at the Mexico defence and beating three markers before placing a left-footed strike into the bottom corner.
Just five minutes later and Martinez doubled his, and Argentina's, tally.
Exequiel Palacios put in a beautiful throughball for Martinez's well-timed run, the forward producing a first-time finish into the bottom corner.
Argentina's incredible start continued when they were awarded a penalty for a Carlos Salcedo handball, the ball hitting the defender's outstretched arm after a Martinez flick.
Paredes stepped up and while Mexico goalkeeper Guillermo Ochoa got a hand on the spot-kick, it found the bottom corner to make it 3-0.
Martinez had Mexico panicking and it showed as he remarkably completed a first-half hat-trick.
The 22-year-old's first touch after a Palacios pass let him down but Nestor Araujo was unable to deal with the loose ball, Martinez claiming possession again before firing past Ochoa.
Martinez came off at half-time and Argentina almost struck again six minutes into the second half, Ochoa saving a strike from Rodrigo De Paul.
Substitutions impacted the rhythm of the contest, although Argentina were never in trouble after Martine's first-half heroics.
What does it mean? Scaloni's Argentina produce one of their best
Scaloni has faced plenty of criticism since taking the helm, but Tuesday's first half was undoubtedly one of Argentina's best since he was named coach in August last year. Martinez's performance helped, but they denied Mexico clear-cut chances and were clinical.
Martinez marvellous for Argentina
The Inter forward had Mexico's defence worried throughout and he took his chances. Palacios' wonderful pass set up Martinez's second, but it was otherwise the former Racing Club doing all the hard work to move onto nine international goals.
Martino unable to stop Mexico's Argentina woe
Under Martino, Mexico have shown they can contend with South American sides, beating Chile, Paraguay, Venezuela and Ecuador during an 11-match unbeaten streak with the former Barcelona boss in charge. But they are now winless in their past 10 meetings with Argentina, a run dating back to the 2004 Copa America.
Argentina will take on Germany in a friendly next month, while Mexico meet Trinidad and Tobago before CONCACAF Nations League clashes with Bermuda and Panama.