By Agence France-Presse
The new legislation permits terminations to be carried out up to 12 weeks of pregnancy -- or in conditions where there is a risk to the life, or of serious harm to the health, of the pregnant woman.
It would also allow terminations in cases of foetal abnormality which could lead to the death of the foetus either before or within 28 days of birth.
"Historic moment for Irish women. Thanks to all who supported," said Varadkar, who supported the referendum in May in which 66 percent voted to overturn a constitutional ban on abortions.
Some 170,000 Irish women have been forced to travel to neighbouring Britain for abortions since 1980.
Ireland is a predominantly Catholic country but the influence of the Church has waned in recent years.
The change means Malta is now the only European Union country to totally ban abortion.
"Just over 200 days ago, you, the people of Ireland, voted to repeal the 8th so we could care for women with compassion," Health Minister Simon Harris said on Twitter after the bill passed the upper house.
"Today we have passed the law to make this a reality. A vote to end lonely journeys, end the stigma and support women's choices in our own country."
The Irish health service is now being primed to provide the first abortions to women in January.
The only step remaining in the legislative process is the ceremonial signing of the bill into law by President Michael D. Higgins.
The vote to repeal the abortion ban followed a vigorously mobilised pro-choice campaign.
"I want to thank the campaigners who fought for 35 years to change a nation, to change hearts and minds," Harris said in a statement.
"I want to thank the minority who fought the battle in here when it was convenient for the majority to ignore."
"We welcome the passage of this bill, and fully appreciate the importance of its enactment by year's end so that abortion services can begin in January," said Colm O Gorman, head of Amnesty International Ireland.
"Women have waited 35 years for this, the daily violations of their human rights must come to an end."
However he said some concerns remained over conditions outlined in the bill -- which has been the subject of sustained and emotional parliamentary debate in recent weeks.