Ms Barker said she was honoured to be "provisionally elected Mayor of Dunedin, for the faith people have shown in me and my vision.
"It’s a wonderful milestone for Dunedin to elect its second woman mayor so I pay tribute to all the women, and men, who paved the way and who supported me during the election campaign."
Ms Barker said she was surprised and excited by the result, although there was "still a sliver in it".
"It could swing, I guess, because there's such a little bit in it. However, we shall just see," she said.
The tight race between Ms Barker and Mr Simms showed the city was in the mood for change, she said.
"There's two front-runners with hardly anything in it and both of us campaigned on change."
If Ms Barker maintains her lead, she would be the second woman elected Dunedin mayor after Sukhi Turner in 1995.
"That's a pretty huge milestone," Ms Barker said.
"I think that would be the most exciting thing for me."
" I had a great team behind me, women and men, and that makes a huge difference is when you have people that believe in you and bore you up through the very challenging election period."
She was sad "to see some experienced councillors look like they have missed out".
Andrew Simms said Barker was ahead of him by about 100 votes with an estimated 10,000 votes still to count.
He expected these to favour Ms Barker.
"I fully congratulate Sophie, if that's the way it plays out, as I certainly think late votes will favour her, so I don't expect the race to change," Mr Simms said.
"I'm obviously disappointed in that respect, but all power to Sophie in doing what she's done, that's great," he said.
Mr Simms said he was "incredibly pleased" to have been elected as a city councillor, along with one member of his Future Dunedin ticket "and potentially another".
"I think our primary concern this morning was that there was a better council, and I think that appears to be the case, and so I think that's good for Dunedin.
"No matter how the rest of it plays out, I think it's in a better place now than it was from a council point of view, and I really look forward to being a part of it."
He said the vote share may not become more clear until "as late as tomorrow".
Jules Radich earlier confirmed he did not win a second term.
Mr Radich said he expected to be part of the council as a councillor.
He had felt during the campaign he was "in the mix", he said.
Mr Radich said in the past term he had shifted the council's focus on to infrastructure.
"It's important that I remain on council because, in the first instance, I want to see the South Dunedin pipes and pump work installed in the ground and done, finally."
