Fianna Fáil leader Micheál Martin has described the byelection as "a good day out" for the party.
They look set to secure the seat in Cork North-Central and the party is polling well in Wexford and Fingal, according to Mr Martin.
Pádraig O'Sullivan looks "destined" to win the seat vacated by party colleague Billy Kelleher earlier this year, he added.
"Historically, Fianna Fail has only won one byelection since 1996 so that gives an illustration of the difficulty of winning byelections. They are hard to win," Mr Martin said.
For us to be clearly destined to win North-Central is a big win for the party. We are very competitive in Wexford and Fingal so it is a good day out for us.
"The byelection is a step on the road to recovery. We had to create a new reality on the ground in Cork North-Central in Padraig. If he didn't make it today, he would have been in a sound position for the General Election."
Mr Martin said that a win today will change the picture for Cork North-Central which, he said, has been poorly represented in the last few years.
"North-Central will have a strong TD which is needed because they feel neglected and people are angry," he said.
Mr Martin said there is "strong disillusionment" with the government and this has been exacerbated by the announcement that Dara Murphy is to resign.
"I was amazed it took so long," he said.
"Dara hasn't been active as a TD for two years since Leo Varadkar did not reappoint him as a Minister. He wasn't acting as a TD here; he was virtually non-existent.
Of all the issues we have had in the last month, that is one of the more serious ones: that someone can not perform for two years and take another job.
"The Taoiseach saying that [Dara] has been very busy is not a satisfactory response. It increases the level is dissatisfaction with politics.
"He should have resigned much earlier. To announce it when he did was bizarre. We could have had two vacancies that could have been filled."