Both sides had impressed with eye-catching football in their respective matches so far in the competition, but a late flurry of goals ultimately booked Mexico a meeting with the Netherlands in the second round.
Few expected Miguel Herrera's men to pose much of a problem for Brazil and Croatia in Group A following a tumultuous campaign in the CONCACAF qualifiers, but their solid defence once again provided a sound foundation.
Croatia had immense trouble creating clear-cut chances, while Mexico threatened with counter-attacks and hit the crossbar through Hector Herrera early on.
The Mexicans were unlucky not to be awarded a penalty with 25 minutes to go following an apparent handball by Darijo Srna, but they were not denied soon after as captain Rafael Marquez headed them into the lead.
Andres Guardado and Javier Hernandez made sure of the triumph with two goals in seven minutes and, although desperate Croatia pulled one back through Ivan Perisic, they were well beaten in the end as Ante Rebic compounded their disappointment with a late red card.
Coach Miguel Herrera was adamant in the build-up that Mexico were not going to simply play for the draw they needed and, although Croatia controlled the early possession, the Central Americans looked typically dangerous on the break.
The encounter's first chance saw Perisic blaze Ivan Rakitic's right-wing cross over in the fifth minute, but Mexico went mightily close just after the quarter-of-an-hour mark as Hector Herrera struck the crossbar with a fierce effort from 30 yards.
Mexico went close again just a few moments later as Giovani dos Santos slipped a clever pass into the danger area for Oribe Peralta, but the Club America forward lost his footing while pulling the trigger.
Croatia continued to look unconvincing in attack as an ill-tempered first half wore on.
The best opportunity Kovac's men could muster was a 31st-minute half-volley from Perisic, which comfortably cleared the bar from just outside the area.
Despite some tactical tweaking from Kovac at the break, with Luka Modric seemingly taking up a more advanced position, Croatia's lack of creative spark was not remedied.
Srna luckily escaped without conceding a penalty in the 65th minute as he appeared to block a Guardado shot with his arms, but it all went wrong for Croatia soon after.
Marquez gave Mexico a deserved lead with 72 minutes on the clock, nodding in Herrera's corner despite Stipe Pletikosa's best efforts and a second swiftly followed.
With Croatia attempting to pile on the pressure, Mexico sprang an incisive breakaway and, after receiving a squared pass from substitute Javier Hernandez, Peralta set up Guardado for an easy finish inside the area.
Mexico gave the scoreline a more resounding look in the 82nd minute as Marquez flicked on a cross and Hernandez was on hand to nod in from close range.
Perisic did get Croatia on the scoresheet with a clever finish from an acute angle, but Rebic ended any hopes of an improbable comeback with a callous challenge on Carlos Pena, earning a straight red as the European side were sent home early.
