Reduced to 10 men in the second half, Costa Rica still seemed to be on course for an upset victory until Sokratis Papastathopoulos scored an injury-time equaliser to force extra time.
Navas, who had blocked a string of shots in the game, was equally impressive in the two 15-minute periods of extra time before he pulled off a superb one-handed save to deny Theofanis Gekas's spot kick during the penalty shoot-out.
Costa Rica went on to secure a 5-3 victory to keep the central Americans' World Cup dream alive.
"To everyone back home and out on the streets - this is for you," said Costa Rica's manager Jorge Luis Pinto.
"The people love football and they deserve this.
We're ambitious, we want to grow and develop and God willing we will go on in this tournament.
Captain Bryan Ruiz scored in the 52nd minute to give the central Americans a 1-0 lead at Arena Pernambuco.
But defender Oscar Duarte was ordered off for a second bookable offence shortly after and Greece poured forward l