Raul Jimenez gave Mexico the lead in the 44th minute before Costa Rica captain Bryan Ruiz levelled from the penalty spot seven minutes after the break.
After a scoreless extra time both teams netted four penalties from their first five kicks.
Carlos Salcedo was successful with the first penalty in sudden death before Ochoa's save from Fuller put Mexico through 5-4.
Mexico will play Haiti, who beat Canada 3-2 earlier on Saturday, in the semi-final in Glendale, Arizona on July 2.
The other two quarter-finals will be decided on Sunday when Jamaica play Panama and defending champions the United States face Curacao.