In the final seconds of a tense match, Danny Galbraith, on for Colin Nish, fired in a dramatic breakaway winner.
O'Dea, instated as captain after a loan deal at Reading, said: "The players need to learn quickly.
"It's not going too well and players will need to step up and be men and win games for the club.
"We can't afford to lose games now. We are 10 points behind and we have to go out with the attitude that we have to win games.
"It's going to be to be tough and we are going to have to do something about it.
"You can't be at a club like this and shirk challenges. We have to roll on to Saturday and win.
"We can't afford to lose any more games. No matter who is on the pitch, it is their responsibility to win the games and not enough players are taking responsibility now.
"We have a game in hand and still have Rangers to play twice so it is not over by any means but it is a big challenge.
"But, as I said, it is up to the players to step up and be men.
"At Celtic you don't get the chance to lose games and mess about. We have to grow up quickly."
Hibs went two points behind Celtic into third place but boss John Hughes would not countenance splitting the Old Firm.
"No chance," he said. "Celtic are still miles out in front of us.
"We are looking for European football. We will hang on in there, we might not make it and will take plenty of knocks along the way.
"We rode our luck, our keeper had some good saves and we defended manfully.
"We played some good stuff in the first half but it was in our own half but there were plenty of positives aside from the result."
Celtic boss Tony Mowbray ignored one question about him feeling under pressure, instead insisting the Parkhead club can still recover to take the title back to the east end of Glasgow.
Asked if Celtic now need a miracle, he snapped: "No, no, no.
"With due respect, that's not a very good question.
"We have still to play Rangers twice and on the evidence of the first two games against them, it's games that we should be looking forward to.
"How many games have we got to go? There are enough games and enough points.
"My message as a coach is that I believe we can win every game, we have to believe that.
"We are building a team and when it clicks it place then let's hope you see us go.
"We had enough chances to score more goals but give Hibs credit, they counter-attacked well late on and scored the goal to win the football match.
"We will keep going. We have to believe in the players and they showed enough evidence that they can win football matches.
"Of course I am concerned about the 10-point gap.
"Celtic fans watch the game, they are disappointed but we are all going in the right direction and when things click in to place then it will be the right direction."