In the letter, the First Minister said that she understood why the recent Brexit chaos at Westminster had left many voters feeling "scunnered" with the political process.
But she claimed such disillusionment was one reason why Remain voters should back the Nationalists at next month's poll to ensure Boris Johnson could not take the country out of the EU.
Ms Sturgeon will continue her tour of marginal seats today in Alloa, where John Nicolson is standing in the Ochil and South Perthshire constituency currently held by Luke Graham of the Conservatives.
But former Scottish secretary David Mundell said that voters faced s face a stark choice between more Nationalist division and the Scottish Conservative plan to deliver for Scotland, former Scottish Secretary David Mundell said today.
Mr Mundell was speaking on the first official day of campaigning alongside Scottish Conservative leader Jackson Carlaw on a visit to Peebles in the Dumfriesshire, Clydesdale and Tweedale constituency.
The text of Ms Sturgeon's letter read: “On December 12 you have a chance to escape from Brexit, and to ensure Scotland’s future is in Scotland’s hands – not Boris Johnson’s or any other UK Prime Minister’s.
“After the last few years I can fully understand if you feel scunnered. Everyone is fed up with this Westminster mess.
“Scotland didn’t vote for this Tory government, and we didn’t vote for Brexit – but the Tories don’t care about Scotland and are dragging us out of the EU against our will. The Westminster system has shown itself to be broken.
“This isn’t some academic debate – Brexit will profoundly affect you, your family and your community. It will harm our economy and our living standards, and it will limit the opportunities of our young people. And it also poses a real threat to public services such as our NHS.
“And under Boris Johnson’s plan Brexit will not be ‘done’ – he hasn’t even started trade talks yet, so we face years of more Brexit wrangling with the very real prospect of a no deal crash-out because of hard Tory demands.
“Scotland is an enormously wealthy country, with human and natural resources that most other countries can only dream of. I believe that our best future would be as an independent country, though I understand that not everyone shares that view. But what I do know is that we can only realise our enormous potential as a nation if we escape from Brexit and decide for ourselves what kind of country we want to be.
“That’s why your vote in this election is so important. If we sit back and do nothing, our future will be decided for us by Boris Johnson and by governments we didn’t vote for.
“We must come together in this election and make Scotland’s voice heard - and the best way to do that is to vote SNP. In every seat in Scotland held by the Tories, it is the SNP who is in second place – so only the SNP can stop the Tories and deprive them of a majority.
“So my message to you is this. Don’t sit back and let the Tories decide your future for you - stand up and be counted. Vote SNP on December 12th to escape Brexit and put Scotland’s future in Scotland’s hands.”
Responding to the contents of the letter, Conservative election candidate Paul Masteron said: This letter from Nicola Sturgeon is desperate, deceitful and duplicitous.
“People who voted Remain did not do so to have their vote co-opted into Nicola Sturgeon’s independence obsession.
“They will see through this brazen attempt to weaponise their vote in the EU referendum to drag Scotland back to another damaging and divisive independence referendum.
“Sturgeon cannot spend the weekend ranting through a megaphone to the party faithful about indyref2 and then expect people to forget it all three days later.
“She has already made clear that she plans to do a backroom deal with Corbyn and force through a Section 30 order before the end of the year.
“Only a vote for the Scottish Conservatives can stop indyref2, end the division, and move on to the things that matter.”