Blue Samurai failed to capitalise on a numerical advantage that they had for 52 minutes of play in Natal following the first-half dismissal of Kostas Katsouranis.
The result leaves Alberto Zaccheroni's side needing to beat already-qualified Colombia next week in order to progress from Group C, and man of the match Honda says that his team will "fight to the end" to achieve that.
"We knew that we needed to win two games," Honda told NTV. "So, it's disappointing that we couldn't get the ball into the goal.
"The result is everything, so regardless of the attacking performance it is disappointing we couldn't get three points.
"Greece played knowing that with a draw and a win in the next game they can go through, and we got caught up in that. We have to fight until the end and not give up."
Japan face Colombia on Tuesday, kickoff for which is scheduled for 9pm.