The game came down to penalties with Mexico scoring five penalties and Costa Rica scoring four.
Mexico will now play Haiti in the semifinals.
"I'm really excited because I'm from Mexico and to be surrounded by nothing but Mexicans, I'm in heaven," said a fan.
The Mexican national team arrived in Houston Thursday, and signed autographs for fans.
The team has yet to lose a game, and many fans believe they'll win the entire tournament with nine straight wins.