Costa Rica captain Bryan Ruiz levelled seven minutes into the second half when he slotted home a penalty and neither team were able to score in extra time.
After both sides netted four penalties from their first five kicks Carlos Salcedo was successful with the first of sudden death before Ochoa's save from Fuller put Mexico through 5-4.
"Mexico played football out there tonight," said coach Gerardo Martino. "We deserved to win because of our football.
"Guillermo Ochoa made the save of the game but everyone was sensational tonight."
Mexico will play Haiti, who beat Canada 3-2 earlier on Saturday, in Glendale, Arizona on July 2 for a place in the final.
The other two quarter-finals will be decided on Sunday when Jamaica play Panama and defending champions the United States face Curacao.
(Reporting by Andrew Downie; Editing by Peter Rutherford)
