There has also been the biggest fall in youth unemployment since 2006, the Scottish Government said, with a drop of 28,000 in the number of 16 to 24-year-olds who are out of work over the last year.
Official statistics showed the overall jobless total - which includes those who are out of work and not eligible for benefits - fell over the the period October to December to 206,000.
That is a drop of 25,000 over the year, with the unemployment rate in Scotland now slightly lower than that of the UK as a whole.
The rate north of the border now stands at 7.7%, compared to 7.8% across the UK.
The youth unemployment rate is also lower in Scotland, at 18.4%, than across the UK where it stands at 20.6%.
The figures for October to December showed that 74,000 young people were out of work north of the border.
Across Scotland the number of people out of work and claiming jobseeker's allowance fell by 600 from December to 137,000 in January - 5,200 less than 12 months ago.
But the number of people in employment also fell by 11,000 over the period October to December to 2,461,000.
The fall in unemployment was welcomed by both Scottish Secretary Michael Moore and Scottish Finance Secretary John Swinney.
Mr Moore said: "In difficult economic circumstances, I welcome the news unemployment fell by 13,000 during the last quarter of 2012.
"It is also encouraging that 5,200 fewer people are claiming jobseeker's allowance compared to one year ago."
He added that the UK Government "continues to work hard to reduce unemployment by laying the foundations for a stronger, more balanced economy".
Mr Swinney said: "This is the third set of monthly unemployment figures in a row that have shown a fall.
"What's more, the fall in youth unemployment is particularly encouraging. Scotland has lower youth unemployment, higher youth employment and lower youth inactivity than the UK. "
He said this month's figures showed the "largest annual drop in the youth unemployment rate since the data series began in 2006".
Mr Swinney continued: "Unemployment fell by 14,000 across the UK as a whole, with Scotland accounting for 13,000 of this net fall.
"But we must not be complacent - too many people are still looking for work and the Scottish Government is taking action to address this by maintaining the most competitive business environment anywhere in the UK and investing in our infrastructure."
Liberal Democrat leader Willie Rennie said the fall in unemployment "can give us hope" but he stressed: "There is no room for complacency given that over 200,000 people remain out of work.
"Both of Scotland's governments must work in hand in hand to provide opportunities for people to return to work."
Mr Rennie added: "The Scottish Government should especially set out its plans to spend the additional £394 million allocated for capital spending from the UK Government's autumn statement."
Liz Cameron, chief executive of the Scottish Chambers of Commerce, said: "It is good news that unemployment in Scotland has continued its downward trend and it is particularly important to note that youth employment fell by 28,000 over the course of last year.
"This has undoubtedly been assisted by the focus on youth employment that has been evident on the part of the Scottish Government in 2012 and Chambers of Commerce across Scotland have also sprung into action to help tackle high levels of youth unemployment."
She added: "Chambers of Commerce are serious about assisting businesses to take on more young people and we will continue to work to deliver real benefits for Scotland's economy and its young people."
Labour finance spokesman Ken Macintosh raised concerns about the fall in employment.
He said: "Whilst the fall in unemployment is welcome, it disguises the very serious and far more worrying upward trend in those withdrawing from the labour market altogether in Scotland.
"Employment is increasing across the UK as a whole, but here in Scotland the number of people in work is falling. 11,000 fewer people are working now than three months ago.
"Women in Scotland are being particular badly affected, with tens of thousands no longer even looking for work."
Mr Macintosh added: "The Scottish Government had an opportunity to present a budget with measures aimed at getting people back into work, but instead we are seeing a further weakening of the jobs market.
"This is a government that has missed opportunity after opportunity to improve the employment prospects of people across Scotland."
Grahame Smith, general secretary of the Scottish Trades Union Congress (STUC), said the figures were "grim reading".
He stated: "Although the headline unemployment rate has fallen quarter on quarter, the reality is that people are leaving the labour force rather than finding jobs.
"Employment has now fallen by 40,000 over the last six months and 55,000 more people have become economically inactive. This situation simply cannot be spun into good news."
He argued that this "prolonged period of stagnant growth, high unemployment, rising underemployment and falling real wages" was the "inevitable consequence" of the UK Government's economic policy.
Mr Smith said: "Unless the Chancellor introduces a targeted, time limited stimulus in his Budget next month there's little prospect of a robust demand led recovery taking hold any time soon."
But Tory finance spokesman Gavin Brown said the Scottish Government needed to focus its efforts more to boost economic growth.
Mr Brown said: "The decreases in unemployment, and particularly youth unemployment, are to be welcomed.
"But clearly unemployment is still far higher than anyone would like it to be.
"More worryingly we've seen another fall in the number of people employed in Scotland, despite there being a large increase across the rest of the UK.
"This illustrates why it is so important for the Scottish Government to focus its efforts more closely on growth."
The Conservative MSP said: "The recent SNP budget did not do enough for the economy, and had particular failings on housing, colleges and taxation.
"It needs to improve its performance, and start matching words with action."