Brenda Hale said her life was turned upside down by a stranger who began harassing her after the 2011 election.
The Lagan Valley MLA said the man started to call and text her, and she felt "incredibly threatened".
"He said we couldn't let this relationship go but I had never met this man," she told the BBC.
She said the man approached her while she was shopping in Marks & Spencer.
"I thought he was a constituent so I was very happy to converse with him and then he said he had been watching me a lot and he had got hold of my telephone number, and then the calls and texts started arriving with increasing regularity," she said.
"I did not know this man from Adam and he was stalking me from his home address in Dublin."
Mrs Hale, whose husband was killed in Afghanistan in 2009, said the whole experience was "terribly frightening".
"I had become a very public person after the death of my husband and so my story was out there - that I was alone with two children," she said.
"To feel vulnerable like that made me question whether a job in the public arena was the right one for me."
She said she "nipped it in the bud very quickly" and the police were "very quick to act".
Northern Ireland is the only part of the UK which does not have specific laws to protect victims of stalking.
On Monday, Mrs Hale brought forward a motion about the need for new legislation. Assembly members voted on it and an amendment introduced by the SDLP and both were passed.
Mrs Hale said the debate was a "vital step forward".
"A legal definition will enable us to not only protect victims but also monitor reports of stalking, arrests and consequent prosecutions," she added.
Green Party deputy leader Clare Bailey MLA spoke of her own experience of stalking after the debate.
"I had razor blades stuck in my car tyres, which caused the tyres to blow up after dropping my children to school," she said.
"I decided then to report the stalking to the police, but response was "What do you want us to do about it?". I couldn't answer their question but I didn't report any further incidences."
It was time to make stalking a "recognised crime in its own right", she added.
Justice Minister Claire Sugden said a review of the existing laws around stalking both in Northern Ireland and elsewhere was already under way.
An assembly member has spoken out for the first time about being stalked, as Stormont backed a motion to safeguard victims.