The two teams meet in the first leg in Tallinn on Friday night and Given recognises the need for Ireland to achieve a positive result.
Ireland narrowly lost out to France via a controversial goal in their 2010 World Cup play-off two years ago and Given is keen to avoid another hard luck story.
"I know we are supposed to be favourites against Estonia but you don't want to be getting carried away," said Given.
"It always gets to this nitty gritty at the end of a campaign. There have been too many sad stories and we want to have a happy one at the end of this. It'll only happen if we do the business on the pitch.
"It doesn't matter what we say or what you guys write, it's all about the performance and getting the result. Please God, hopefully we can do that."
The 35-year-old goalkeeper has recovered from a neck injury and is fit for Friday's match.
Given joined Aston Villa from Manchester City during the summer transfer window.