The Greeks repelled wave after wave of Japan attack as the Blue Samurai failed to capitalise on their numerical advantage following Kostas Katsouranis's red card after 38 minutes.
Zaccheroni's side must now beat already-qualified Colombia in their final Group C game to stand any hope of progressing into the last-16 stage, and the Italian says that they will be going for the win on Tuesday.
"We are disappointed, we were hoping to win," Zaccheroni told reporters. "The circumstances were there but the knockout blow was missing.
"We created some chances without managing to take them, even though I was hoping for more goalscoring chances. We lacked that sharpness in the penalty area.
"The last match? We'll play to win, as always."
Japan lost their opening fixture to Ivory Coast on Sunday morning.