The match ended 1-1 through extra time.
Raul Jimenez scored in the 44th minute to give Mexico the lead before halftime.
Bryan Ruiz tied the score for 39th-ranked Costa Rica in the 52nd minute with a penalty kick after he was taken down by Luis Alfonso Rodríguez.
The 18th-ranked Mexicans will face 101st-ranked Haiti on Tuesday at Glendale, Arizona.
Source: FOX Sport