All those 16 and over who have not recorded a decision about donation will now be considered a possible donor, in consultation with their families.
It comes after similar law changes in Wales and England in recent years.
Dr Sharon Zahra, a consultant in tissues and cells transfusion at the Scottish National Blood Transfusion Service, said the law change will save and improve lives.
It is set to increase the number of people on the donor register, but donations will always be made in consultation with family members and after a system of “checks and balances”, Dr Zahra said.
"The aim is to raise awareness about the importance of donation,” she said.
"To make it the norm that we discuss this as community, that you have had a conversation with your family about what you would like to happen if the worst were to happen. If donation were to be possible, would you want to be a donor or not.”
She added: “The vast majority, if not all, of our donors are sudden and unexpected deaths.
"It's an incredibly traumatic time for anyone to deal with, and I think if your family already know what you would have wanted to do, because it's become normal for us to talk about it, it's become normal for us to register what we want or would not want to happen, then their decision at the time to support your wishes is made much easier because they know what they are.”
Dr Zahra stressed that the law change will affect tissue as well as organ donation.
Tissue donations, which include life-saving things such as heart valves and skin, and life-enhancing donations such as tendons, can be made up to 48 hours after death, so in some cases are feasible when organ donation would not be.
Elaine Kennedy, 45, from Uddingston, lost her husband David in March 2019.
Mr Kennedy, 43, died suddenly after an accident at work, and went on to donate his kidneys, liver and pancreas.
Ms Kennedy was aware of her husband’s wish to be an organ donor after a chance conversation when filling in a form, and said she hopes the law change will allow other family members to have important discussions about organ donation.
“If I didn’t know what he wanted, it would’ve made it harder. Because of that conversation, myself and David’s family were able to honour his decision,” she said.
"I had more time to digest David’s wishes than his family and I was clear about what he wanted to happen when I raised it with the doctors. But after talking it through with them, in that moment, we all wanted to respect David's wishes.
"He was such a selfless and generous person in life, he wanted to make everyone happy and this was the last thing he could do for someone else.”
She added: "It’s bittersweet – you never want to find yourself in that position, but the fact that there’s three recipients and families out there who may have been given a quality of life they didn’t have before, has helped in our devastation.”