Youth crime fell by 22% in the last year and anti-social behaviour was also down, said a report into police performance.
Justice Secretary Kenny MacAskill said the figures were "encouraging".
The Scottish Conservatives accused the government of "crowing".
Crime in Scotland is now at a 39-year low, according to the Scottish government.
Almost 810,000 crimes and offences were recorded in 2012-13, 5.7% fewer than in the previous year.
Youth crime saw the biggest fall, down 22% since 2011-12 and 52% since 2008-09.
Anti-social behaviour was down by 13% on 2011-12, and racially motivated crimes and offences fell 21% in the same period.
Police detection rates rose slightly over the last year.
The report also showed that 3,314 sex offenders were living in communities in Scotland, an increase of 3% on the previous year and 330 more than in 2008-09.
Public satisfaction with the police was high, at 84%, although only half of people were confident that the police could prevent crime, and only 9% said they felt safer in their community.
The Scottish Policing Performance Framework Annual Report covers the period immediately before the formation of a single police force for Scotland in April this year, and breaks down crime and detection rates for the previous eight regional forces, as well as calculating national averages.
Mr MacAskill said: "The report shows that police are continuing to exceed expectations while working hard to cut crime and keep our communities safe.
"It is especially encouraging to see significant decreases in crimes which can have a negative impact on communities such as youth offending - down by over half since 2008-09.
"This is backed by the lowest rates of recorded crime in almost 40 years and over 1,000 extra officers in communities, coming at a time of the biggest changes to policing for a generation."
Police salaries totalled Â£897.8 million, up 1% on 2011-12.
Labour MSP Graeme Pearson said: "While these figures show welcome progress, there are significant challenges remaining for Police Scotland, particularly around the confidence people have in the police to prevent crime and to catch those responsible.
"SNP cuts mean our thin blue line is getting thinner. Police stations are closing to the public, police officers are being taken off our street to cover back office cuts and more cuts are in the pipeline.
"With budget cuts of almost Â£140m over the next two years, how Police Scotland remains local, responsive, accessible and flexible will be the biggest challenge."
He added: "It is simply unacceptable for the SNP to dismiss all the difficult decisions facing the police as 'operational matters' whilst wishing to simultaneously claim credit when the statistics are moving in the right direction."
Conservative justice spokeswoman Margaret Mitchell MSP said: "This report shows, despite constant crowing from the Scottish government about its performance on crime, the public has a very different perception.
"The fact only half the people in Scotland have faith in the police to do the very job we need them to in preventing crime, coupled with only 9% believing their local community is safer, must surely be a matter of real concern to both the Scottish government and Police Scotland.
"More worryingly , this report reveals that a record number of sex offenders are now living in our communities. And even more concerning is the fact that a high number are breaching their licence conditions and ending up back in jail."
Police Scotland Deputy Chief Constable Rose Fitzpatrick said: "The service is already performing well at a national level in its first year, with violent crime down and robust action against organised crime groups. This strong performance can also be seen at a local level, with reductions in anti-social behaviour and disorder.
"This is a year of changing processes and new ways of working against a backdrop of financial savings, so the performance to date is encouraging."
Vic Emery, chairman of the Scottish Police Authority, which maintains policing in Scotland and holds the chief constable to account, said: "Scrutinising the results of how police time and resources are used is vital so that we can ensure that police priorities and performance are aligned, and that we continue to test the quality of policing outcomes against the necessary reductions in cost that must be made."
Scottish police are "exceeding expectations" as crime continues to fall, according to figures released by the Scottish government.