The Labour frontbencher insisted Scots needed to come together if the country was to move forward in the wake of the vote and vowed not to rest until the people received all they had been promised.
She was speaking at the party’s annual conference in Manchester where she was joined on stage by Labour leader Ed Miliband.
He used the occasion to thank all those instrumental in securing a No victory, but left out Gordon Brown from the list.
The former prime minister’s efforts had previously been acknowledged by Ms Curran who called him a man who “electrified this campaign”.
Turning to the referendum result, she said: “We have to be honest when we look at the results and see that some of the people who think that Britain can’t work for them are Labour voters.
“They are terrified of Tory governments, they are worried about the future of our public services, and they are looking at their living standards which have been ground down year on year.
“We need to understand why they are angry and what we need to do about it.
“We need to understand why, in areas like Glasgow, Lanarkshire and Dundee people feel so let down that they want to opt out entirely.
“While the decision of the Scottish people is absolutely clear, I understand that many of my fellow citizens are hurting and they believe that progress on their hopes for Scotland has stalled.
“Our most important task now is to persuade them they should not be afraid.
“We may have disagreed on the means, but we share a very similar vision about the kind of Scotland we want to build.
“And that vision can still be realised. So if you voted Yes because you wanted a fairer and better Scotland, then we can work together.
“If you voted Yes because poverty and inequality cuts you to the core, then come with us.
“And if you voted Yes because you want to build a movement to change our country, I say the Scottish Labour Party is not your opponent but your home.”
Scottish Labour leader Johann Lamont echoed her call saying the Scottish Parliament must be used to address a “deficit of hope”.
“Because at the end of the day powers to parliaments mean nothing if they don’t mean the empowerment of people,” she added.
“I believe those who sought change through separation were wrong, but I salute their passion for change, I salute their commitment and I ask them to share their energy with us to change Scotland and change Britain and build that society we all seek without borders.
“There is only one goal no matter how the paths may differ. The goal is a truly just society where the talents of all can flourish and the talents of none is wasted.”
At the end of the speeches, Mr Miliband - who joked that his conference speech had not been brought forward 24 hours - welcomed colleagues stage for a celebratory group photo.