Vote-counting began at 0800 GMT across the country on Saturday, with the first offiicial indication of results expected mid-morning.
The anti-abortion campaign, which wanted to keep the Irish constitution's eighth amendment, seemed prepared to accept a heavy defeat but vowed to stand firm.
In light of the exit poll results, the main group opposing the liberalization of the country's abortion law conceeded on Saturday that it had lost the election. Halappanavar, who was 17 weeks pregnant at the time, had asked for an abortion after experiencing severe back pain and was miscarrying but the doctors refused because of the country's law. "We've done it", said Kevin Humphries, a former junior minister who campaigned for more access to abortion.
PAAn exit poll is a poll of voters taken as they leave polling stations
The Save the Eighth group said: "What Irish voters did yesterday is a tragedy of historic proportions".
#RepealThe8th, promoted by those who are pro-choice (against abortion ban), has been used nearly 90,000 times on Instagram, and even more times on Twitter and Facebook.
The referendum comes three months before a visit by Pope Francis for the World Meeting of Families (WMF) and three years after Ireland voted to legalise same-sex marriage despite the Church's opposition. Currently, terminations are only allowed when a woman's life is at risk.
The RTE poll suggested the highest "Yes" vote was in Dublin, where 80 percent were in favour, but there was no sharp urban/rural divide as in previous referendums on the subject, with 63 percent of people living in areas with a population under 1,500 backing the proposals.
For advocates of repeal, a "yes" vote would be a landmark in Irish women's fight for equality and the right to control their own bodies.
A voter arrives at a polling station in Dublin.
"Yes" campaigners have argued that with over 3,000 women traveling to Britain each year for terminations - a right enshrined in a 1992 referendum - and others ordering pills illegally online, abortion is already a reality in Ireland.
The Irish Times and RTE television exit polls suggest the Irish people have voted overwhelmingly to repeal a 1983 constitutional amendment that requires authorities to treat a fetus and its mother as equals under the law.
Abortion was illegal in Ireland under the Offences against the Person Act of 1861, but when contraception was legalised in 1974, there was concern from many conservatives that abortions would follow.
Mr Varadkar claimed Ireland was united - with men and women, almost every age group and every social class opting for reform in Friday's referendum.
The vote saw citizens effectively opt to either retain or repeal the Eighth Amendment of the state's constitution, which prohibits terminations unless a mother's life is in danger.