Health Secretary Nicola Sturgeon launched a Government-commissioned study into the measure today and heard first-hand accounts of the impact of alcohol abuse during a visit to Edinburgh Royal Infirmary.
The study by Sheffield University academics suggests that hundreds of lives can be saved every year with minimum pricing, which may also reduce crime and alcohol-related illnesses.
And over 10 years 950 million can be saved in health spending, criminal justice and employment if a 40-pence-per-unit minimum price was implemented.
Ms Sturgeon said: "Alcohol misuse, if you take into account the health impact, the economic impact, the crime impact, costs us 2.25 billion a year.
"So 1 billion over ten years is realistic and it's going to make a big difference.
"And that's before you look at the human costs and the lives that are going to be saved, as well as the reduction in the number of people getting ill."
Ministers have not yet said what the proposed minimum price will be but 40p per unit has been used as an illustration.
This figure, coupled with a ban on promotions, will see alcohol-related deaths fall by about 70 in the first year of the policy, then by about 365
after 10 years, according to the study.
A minimum alcohol price will also lead to 30,000 fewer absence days from work a year, the study said.
Alcohol-related illnesses will also fall by 1,200 in the first year and 3,700 for every year thereafter, once the policy has been running for ten years, the study said.
Crime is also expected to fall by 3,200 offences a year.
The greatest impact of minimum pricing should be on heavy drinkers who tend to buy cheaper, high-strength products such as white cider.
Moderate drinkers will barely be affected at all, the study said.
The SNP minority Government will need opposition support to get the measure through Parliament in a forthcoming Bill.
The Health Secretary said she believes some Labour MSPs are prepared to "look seriously" at the proposal.
"I think that's the same in other parties as well," Ms Sturgeon said.
"We'll continue to make the case, continue to try to build a consensus."
The proposed Bill will also include a ban on off-sales promotions and a "social responsibility fee" for some retailers.
A Labour spokesman said today: "We will look seriously at any sensible ideas from any source that will reduce the level of problem drinking in Scotland but we are still waiting for the SNP to bring forward a credible package of proposals."
Ms Sturgeon watched a presentation at Edinburgh Royal Infirmary from Dr Alastair MacGilchrist, who warned about the growing number of people with liver damage because of alcohol abuse.
"The death rate from cirrhosis is doubling every decade in Scotland," he said.
"In the 30 years since I qualified, I've seen a dramatic increase.
"At a time when the death rates from cirrhosis are falling, they're rising in the UK and they're rising faster in Scotland than they are in England."
Dr Brian Keighley, chairman of British Medical Association Scotland, welcomed the proposal today.
"It is clear from this evidence that the introduction of a minimum price, as part of a wider strategy, will be a significant step towards improving the health of Scots."