Green Party leader Clare Bailey outlined her experience as a volunteer at the Marie Stopes clinic in Belfast city centre as she opened a debate on the proposed law changes.
Her private member’s bill later passed its latest assembly stage by 58 votes to 29.
It will now proceed to the committee stage of the legislative process.
Northern Ireland’s restrictive abortion laws were liberalised in 2019, but provision of services has yet to be fully rolled out across the region amid a political dispute over the highly sensitive issue.
South Belfast MLA Ms Bailey said a “campaign of harassment and intimidation” outside facilities offering abortions or family planning advice had “escalated” in recent months.
However, she said her desire for exclusion zones pre-dated recent events.
The Marie Stopes clinic, which closed in 2017, was the scene of regular protests by anti-abortion campaigners.
When volunteering with Marie Stopes, she said, “I was spat at, I was assaulted, I had holy water splashed on me, I was verbally abused, I had one young woman who was so distressed that she ran into four lanes of oncoming traffic to try and escape the protesters.
“I had another young woman alone in the city being filmed and threatened to be uploaded and broadcast on social media.”
The DUP said it would vote against Ms Bailey’s Bill.
Party MLA Jonathan Buckley expressed concern at the potential limitation on the right to protest if exclusion zones were introduced.
He said his party opposed any form of abuse or harassment but said the bill would constrain anti-abortion campaigners who wanted to lawfully express their views.
“There is a real risk, members, that the clauses which constitute the main fabric of the Bill would unfairly restrict freedom of assembly, expression and religious belief, as set out in articles 9, 10 and 11 of the European Convention on Human Rights,” he said.
If the Bill passes the second stage it will be passed to an assembly committee to undertake detailed scrutiny of its proposed provisions.
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