There were 54 deceased donors in 2007/08, rising to 94 in 2012/13, the NHS blood and transplant service said.
Meanwhile, the number of people on the NHS Organ Donor Register in Scotland has continued to rise, reaching a record high of more than two million, or 41.1% of the population - the highest of any UK country.
The statistics show that Scotland has surpassed the target set by the Organ Donation Taskforce in 2008 to increase the number of donors by 50% over five years.
A total of 285 people have received life-saving transplants in the past 12 months while the number of deaths on the waiting list dropped from 55 to 34 over the five-year period.
Public health minister Michael Matheson said: "I am delighted that Scotland has delivered a 74% increase in organ donation and I would like to thank the many NHS staff who have worked tirelessly to make this possible.
"I am very aware that organ donation can only occur as a result of tragic circumstances but I also know that many donor families gain comfort from the fact that their loved one went on to save several lives.
"Reaching this increase is a landmark achievement, however we must not forget the fact that around 600 Scots are still waiting for a transplant and we can and must do more to help them.
"We will continue to work with NHS Blood and Transplant and the other UK countries to build on our achievements and the excellent progress which has been made to date."
Professor John Forsythe, Scotland's lead clinician for organ donation and transplantation, said: "This is a landmark achievement for Scotland and the UK as a whole.
"When we embarked on this journey five years ago we knew it would be difficult to achieve the challenge set by the Taskforce.
"However with significant improvements right across the field of donation and transplantation we have increased the number of donations in Scotland by over 74% and seen a significant rise in the number of transplants."
Dr Sue Robertson, a renal physician and member of the British Medical Association's Scottish Council, said the organisation is "delighted" at the increase in donors.
"The whole transplant community has worked tremendously hard to achieve this with significant support from the Scottish Government," she said.
"However, we agree with the chair of the Organ Donation Taskforce, Elisabeth Buggins, that more can be done and more lives can be saved. "
"Last year the BMA issued a report, Building on Progress: Where next for organ donation policy in the UK? that considered other options to increase organ donation.
"The report concluded that the best way forward for the UK was to introduce an opt-out system with safeguards, otherwise known as a 'soft opt-out system'.
"Unless an individual had registered an objection to donating their organs, or their family was aware of any objection, the default position would be to donate. Creating this kind of system would allow the wishes of the individual to be taken into account and would increase further the number of organs available for transplant."