The charity NSPCC said there were more than 3,000 crimes including rape, sexual assault and grooming recorded by police in 2013-14.
Publishing its annual How Safe Are Our Children? report, the charity said it had seen a 19 per cent rise in the number of adults contacting its helpline to report concerns about a child being sexually abused during the past year.
Half of the contacts received were so serious that they were immediately referred on to the police or children’s services.
The NSPCC said the growing number of victims coming forward meant it was time to reform the legal system, making it easier for children to give evidence against their abusers.
Using figures from the Scottish Government, the NSPCC said there were 3,742 recorded sexual offences against those under the age of 18 and 3,101 against those under the age of 16 in 2013-14.
The figures include nearly 100 rapes or attempted rapes of children under the age of 13 and 349 sexual assaults of children under that age. Figures from England and Wales show there were 31,238 allegations of sexual offences against children, including rape and assault in 2013-14, an increase of more than a third compared to the previous year.
Matt Forde, NSPCC Scotland’s head of national services, said high-profile abuse scandals had helped increase public awareness of the issue.
He said: “It’s not clear from the raw numbers, but we think it’s very likely a greater awareness plays a part (in the rise).
“We’ve seen a spike in inquiries and calls to our helpline since revelations about Jimmy Savile and the other cases that have emerged. It has triggered people who have kept quiet for many years and given them the confidence to come forward as well as more broadly raising awareness in the public.”
Mr Forde said cases such as the Rotherham abuse scandal had raised “real concerns” about vulnerable young people being groomed for sex.
Asked if a similar abuse scandal could happen in Scotland, he said: “There’s no room for us to be complacent about that. We need to be alert and vigilant as what was uncovered in Rotherham was shocking.”
The NSPCC said the legal system had to develop a more “sensitive and effective” response to victims. It said multiple interviews in police stations and hospital could “re-victimise” the child, triggering memories of their abuse.
Kath Harper, head of the Crown Office’s National Sexual Crimes Unit, said: “Raised awareness of sexual offending against children and an increased confidence in police and prosecutors will undoubtedly play a part in the higher number of reported sexual crimes against children.
“Such offences are a high priority for the prosecution service which is why we have appointed specialist prosecutors and have had a dedicated National Sexual Crimes Unit since 2009.
“This unit means that the Crown is ideally placed to deal effectively and sensitively with cases of child abuse.”
A Scottish Government spokesperson said: “Any offence of this nature towards children is appalling and while the report highlights increased levels of reporting, we know that a single case is one too many.
“With police and prosecutors we take these traumatic crimes extremely seriously.
“That’s why we have strengthened the law, established specialist prosecutors through the National Sex Crimes Unit and police have improved investigation techniques, setting up the National Rape Taskforce and National Child Abuse Investigation Unit.”