Giovanni Trapattoni's men will run out at the A Le Coq Arena in Tallinn knowing a draw would be a decent result to take back to the Aviva Stadium on Tuesday evening.
However, Keane is adamant that a win will be the only thing on their minds as they attempt to take a major stride towards qualification for next summer's showcase finals.
He said: "Listen, as players, you don't go into a game thinking you are going to draw, it's impossible to do it.
"You go into every game thinking you are going to win it, so we are going into this game thinking that we can win it because there is no reason why we can't.
"We have the players to do it and we have the mentality to do it, so there's no reason why we can't do it.
"We go into every game with the same mentality, that we can win this and we can score goals."
The Republic's recent run of form backs up Keane's claims - they are unbeaten in nine games and have conceded just one goal in the process.
Keane is confident that is evidence of the improvement they have made under Trapattoni's guidance in the last three and a half years or so.
He said: "You have seen over the years since the manager has come in, we have grown as a team, we have got better with each game and that reflects in our performances and the fact that we haven't lost too many games.
"Of course, we are a better team. The players are getting more experienced and able to handle different situations, so we are certainly a better team than we were a couple of years ago."
At one point, the chances of Keane playing in tonight's game looked slim when he limped away from the 2-0 qualifying victory in Andorra with an adductor muscle injury.
Even the man himself had his doubts, but a swift recovery and three games in a week for club Los Angeles Galaxy have seen him return to full fitness in the nick of time.
Keane said: "Initially when it first happened and the doctors told me it was four to six weeks, of course there was a little bit of doubt.
"But as soon as a few days go by, you have to mentally get yourself right.
"Since I have been a player, if people say four to six weeks, I always try to take a couple of weeks off mentally in my own brain, and that's exactly what I did with this one.
"I had good people behind me at the Galaxy - the physio, Armando (Rivas, head athletic trainer), was brilliant and he has helped me to get fit a lot quicker."