The Mexicans were frustrated by the plucky Caribbeans for 84 minutes, but Raul Jimenez slotted a goal to break the deadlock from a Miguel Layun cross.
The match-winner was far from clinical, coming from a streaky kick that snuck into the bottom left corner of the net.
"We played at their style with their rhythm. They wanted to be powerful but we were able to control it and win it at the end."
Orozco said Mexico would be ready for Panama in Texas on Wednesday, having earlier suffered a 2-1 defeat to the central Americans at the pool stage.
"We have a chip on our shoulder and we are going to take revenge against the Panama side," he said.
"We have a very solid goal and we are going to fulfil it all the way to the end."
Earlier, Panama were too good for Cuba despite trailing early when Jose Ciprian Alfonso scored in the 21st minute off a volley.
Gabriel Torres scored twice, including a penalty, in the first half as Panama took a 2-1 lead at the break.
His fellow striker Blas Perez also netted two goals, both in the second half, while defender Carlos Rodriguez and substitute Jairo Jimenez each scored once.
Cuba only scraped into the quarter-finals after an injury time goal in their last group match and their hopes of pulling off an upset effectively ended when Ariel Gonzales was sent off in the 58th minute for a clumsy challenge.
The other two quarter-finals will be held in Baltimore on Sunday with the winners also advancing to Wednesday's semi-finals at the Dallas Cowboys Stadium.