Sinn Fein President Gerry Adams has topped the poll in the Louth constituency, winning a seat on the first count.
He won 15,072 votes easily surpassing the quota of 13,864. Fine Gael's Fergus O'Dowd came second, winning 13,980.
Sinn Fein has now won five seats in the Irish general election.
Mr Adams announced in November that he had decided to step down as MP for West Belfast and stand for election in the Irish Republic.
He said that he would be a "strong voice" for working people in the Irish parliament.
"We went out in this election, we set out our stall very, very clearly," he said.
"I think the votes across the state show a significant amount of people support the position we have taken up."
After criticism of his performance in debating economic issues, he dismissed claims that this might resurface in the Dail.
"I don't have any concerns at all about that," he said.
"The people who are challenging me on our economic position are the people who brought the economy to its knees."
He refused to be drawn on whether he would lead the party in parliament.
He added: "It is a huge privilege and I will give it 100% in whatever capacity I represent them (Sinn Fein) in the Dail."
Earlier, he congratulated Fine Gael leader Enda Kenny on the prospect of becoming taoiseach (prime minister).
"If Enda becomes taoiseach, I wish him well," Mr Adams said.
"We will support him when he is doing things that we think are good and progressive and we will oppose him tooth and nail when he is doing things that are not in the common good."
Sinn Fein has now won five seats in the Dail, with Pearse Doherty and Caoimhghin O Caolain topping the polls in Donegal SW and Cavan Monaghan respectively.
Mr Doherty won 14,262 first preference votes while Mr O Caolain won 11,913.
The Cavan Monaghan candidate won 33 votes more than the quota.
Sean Crowe won in Dublin South West while Padraig MacLochlainn took a seat in Donegal North East.
Overall, Sinn Fein is expected to comfortably record its best performance in an Irish general election in the modern era.