Although Costa Rica played the final 24 minutes of regulation plus extra time with 10 players following Oscar Duarte's red card, Greece failed to overrun the Central Americans in the additional half hour in Recife before losing the penalty shootout 5-3.
Greece, who made it 1-1 in second-half stoppage time, missed a number of big chances in extra time, wasting a counter-attack when five players surged against just two Costa Rican defenders, while Kostas Mitroglou mis-hit a late shot, allowing Keylor Navas to save.
Costa Rica were worn out when the 120 minutes ended but Santos claimed Greece were in a similar position and that was why they could not break down the undermanned CONCACAF side.
"Of course in extra time we had no organisation and our minds were not sharp due to exhaustion," the Portuguese coach said.
"We tried to push every way we could to score another goal but in the end it went to a shoot-out."
With both teams aiming for a maiden quarter-final appearance at a FIFA World Cup, Bryan Ruiz gave Costa Rica the advantage in the 52nd minute with a finish from the edge of the box.
Greece then gained control of the match after Duarte was sent off and equalised in the first minute of second-half stoppage time when Sokratis Papastathopoulos converted a rebound after Theofanis Gekas' initial shot was saved.
In the shootout, Navas - who was named man of the match by FIFA - saved Greece's fourth spot-kick from Gekas before Michael Umana scored the winner for Costa Rica.
"Navas was brilliant," Santos said.
"Without him, the outcome would have been different."
Santos was sent off during the break before the shootout and the 59-year-old stubbornly argued the point with referee Ben Williams before eventually trudging into the bowels of the stadium.
After the match, Santos remained bemused as to why he had been expelled.
"It was inexplicable," he said.
"I can't figure out why. I said something very simple to the referee, that I had walked on to the pitch after the final whistle of extra time to support my players.
"I saw the Costa Ricans did the same but I was punished and sent to the stands."
