Pro-choice campaigners have hailed the Yes vote as a resounding roar for "dignity and decency".
Politicians and supporters of repealing the Eighth Amendment said it was a huge step forward for Ireland.
They made the comments as final tallies at count centres across Ireland indicate almost 70% of Irish people have voted to liberalise the country's strict abortion laws.
Together for Yes co-director Orla O'Connor said the exit poll results showed "a resounding roar from the Irish people" for repealing the Eighth Amendment.
"Our campaign and we will be forever indebted to those women and couples whose own bravery and dignity have moved hearts and changed minds - and given the scale of the victory, changed the country," Ms O'Connor said.
Co-director Ailbhe Smyth said: "This will be a moment of profound change in Ireland's social history, a moment when the nation collectively stood up for women and for their healthcare, and voted for constitutional change.
"Together For Yes always knew that Ireland was ready for this change, because of the evidence and facts showing the harm and the pain of the Eighth Amendment."
Irish deputy premier Simon Coveney described it as a vote to put a "more compassionate and appropriate" policy in place, one that helped and respected women in vulnerable circumstances.
"This is a huge step forward for Ireland," Mr Coveney said.
Health minister Simon Harris said he could not have predicted the scale of the victory.
"Under the Eighth Amendment, the only thing we could say to women was take a flight or take a boat and now the country is saying no, take our hand, we want to support you," Mr Harris said.
"Women in crisis pregnancy, we were telling them or letting them feel like they were on their own - and today the country is saying no, we want to stand with you."
Amnesty Ireland executive director Colm O'Gorman said the outcome would be a huge milestone for women's rights.
Mr O'Gorman said, by voting resoundingly to end the constitutional ban on abortion, Ireland was sending a powerful message to women and girls in Ireland and across the globe.
"This is such an important vote for women's dignity and bodily autonomy," Mr O'Gorman said.
Vocal Yes campaigner Dr Peter Boylan described the result as a "watershed moment".
The Institute of Obstetricians and Gynaecologists chairman said he was "very relieved" and felt "vindicated".
"It's a wonderful day for Irish women," Dr Boylan said.
He paid tribute to the women who "told their stories so bravely".
Dr Boylan also thanked his the consultants at the coalface, who he said witnessed at first hand the reality the Eighth Amendment made.
Children's Minister Katherine Zappone said she felt emotional and expressed deep gratitude to voters.
She pledged the legislation would be introduced quickly, before the end of the year.
Senator Catherine Noone, who chaired a parliamentary committee which recommended the abortion changes, said she was very emotional when she heard the predicted results.
Ms Noone said the resounding majority had surpassed her expectations.
She added it showed that many politicians were "somewhat out of touch" with what the people think on the ground.
Opposition TD Micheal Martin told RTE News that the Irish people had made the right decision and it would mean better care for women in Irish hospitals.
"It's the dawn of a new era," the Fianna Fail leader said.
He said he wrestled with the topic, but that women who told their stories had a big impact on him, especially those who had suffered rape, incest and fatal foetal abnormalities.
Sinn Fein president Mary Lou McDonald said it was an important message of compassion and progress and the vote was "cathartic".
She added: "We have without doubt done right by Irish women for this generation and many to come."
She said there was no reason to delay legislation.