Bryan Ruiz put Jorge Luis Pinto's side ahead but they were reduced to 10 men when Oscar Duarte was sent off. Santos's side managed to find a stoppage-time equaliser through Sokratis Papastathopoulos to take the game into extra time and then penalties.
Santos told reporters: "I don't like talking about good luck or bad luck. Someone has to be lucky and tonight it was them, and I congratulate them for that.
"I think Costa Rica were the better side in the opening 15 minutes, but then we came back into it. We had to go and take more risks after the goal, but they defended well and tried to hit us on the counter.
"In the end, though, we managed to take the game into extra time, when we put pressure on them but weren't sharp enough, which is perfectly understandable when you're tired and tense. Navas was brilliant. Without him, the outcome would have been different."
Costa Rica face the Netherlands in their last-eight clash later this week.