The Republic have managed 10 points from four games but skipper Keane admits they must believe they can reach South Africa.
He said: “Belief comes with results. When you are winning games, that brings belief into the squad — the belief that we can go on and win games.
“You can see that over the last four games. As long as we believe we can win, no matter who we are playing, we’ll do all right.
“Against Georgia, coming from behind helped. We’re usually conceding in the last minute of games and usually struggling then.
“But we had time to come out in the second half and play well. Now we’ve got the belief that we can come back any time.”
He added: “International football is the biggest honour.
“You can ask any player in our squad, ask any player in any other country’s squad.
“As a kid, you grow up dreaming of going out and playing for your country.
“But I think Irish players are more proud. And that serves us well.”