She said those "appalled" at the behaviour of the Westminster Government should move to Scotland and be part of an "outward-looking, compassionate country".
A stand-off is growing between the Scottish First Minister and Prime Minister Theresa May over the prospect of a second independence referendum.
The SNP leader used her closing address to her party's conference to highlight the difference between the two administrations.
An independent Scotland would "unequivocally" guarantee the right of all EU citizens living in the country to stay here, the First Minister said.
She also used her speech in Aberdeen to highlight the "massive opportunities" Scotland has as a country "if we choose to grasp them".
On the issue of a second independence referendum, the First Minister said she would negotiate "within reason" on the timing of such a ballot with the Prime Minister.
But she stressed that if - as expected - MSPs back her request for Westminster to transfer the legal power to allow Holyrood to hold another constitutional ballot, then this would become the will of the Scottish Parliament.
She insisted: "The will of our parliament must and will prevail."
With Mrs May currently preparing to trigger Article 50 to start the formal Brexit process, Ms Sturgeon said the UK was "about to turn its back on membership of the world's biggest single market".
She told the conference: "Imagine what will happen if Scotland chooses to stay.
"We will become a magnet for talent and investment from all across the UK.
"So let me issue this open invitation today - Scotland isn't full up.
"If you are as appalled as we are at the path this Westminster Government is taking, come and join us.
"Come here to live, work, invest or study.
"Come to Scotland - and be part of building a modern, progressive, outward-looking, compassionate country."
She hit out at the Tories over their attitude to Scotland - saying the Prime Minister's "refusal to budge an inch" in Brexit talks between the governments had forced her to announce plans for a second independence referendum.
Ms Sturgeon made clear that "there will be an independence referendum", telling party activists that "Scotland's future must be Scotland's choice".
She added: "To stand in the way of a referendum would deny us that choice.
"It would mean that the path of our country was determined, not by us, but for us. Decided by an increasingly right wing, Brexit-obsessed Tory Government."
She accused the Conservatives of harking after the days "when Tory governments could do anything they wanted to Scotland... the days when they could impose the poll tax, destroy Scottish industry and deny all demands for constitutional change".
But the First Minister told Mrs May: "Those days are gone and they are not coming back."
She warned the Prime Minister that if she showed "the same condescension and inflexibility, the same tin ear" to other European nations as she had done to Scotland then "the Brexit process will hit the rocks".
The SNP leader blasted the UK Government over its failure to guarantee the rights of EU citizens already living in the UK.
"You cannot lecture others about politics not being a game while you are using the lives of human beings as pawns," she told Mrs May.
Additional reporting by Press Association