Jose Holebas' first-half strike ensured Giovanni Trapattoni's side finished the year without victory, despite showing plenty of endeavour.
The hosts missed a number of good opportunities, most notably through Simon Cox, as well as strong claims for a penalty when Konstantinos Stafylidis looked to have handled Stephen Ward's cross.
"I thought we played very well, I thought we deserved to win the game. We passed the ball very well at times and we're just disappointed to get nothing from it," Coleman told Sky Sports.
"I thought we definitely had a penalty claim in the first minute of the game. I thought it was a stonewall penalty but I thought we played all right. Just disappointed to get nothing from it."
Meanwhile, O'Shea admitted he should have done better but was pleased with the team's overall performance.
"It's a step forward but we're obviously disappointed," he said. "For the goal I should have done better, I let him come off just too easy but I think we should have been a couple of goals up by then.
"We had the penalty shout early on and the way we started the game we probably deserved to be ahead but credit to Greece.
"They haven't been beaten in a few games and they've put in a determined, defensive performance.
"In the second half we probably could have done things a little better quicker and better in the final third to create more clear cut chances but we had the confidence to knock it about and anytime we got Seamus on the ball down the flanks, we looked very dangerous."