The airport was put up for sale by its owners, New Zealand-based firm Infratil, in March 2012, and is currently running at an annual loss of £2 million.
Deputy First Minister Nicola Sturgeon told MSPs in a statement that ministers hope to complete negotiations with Infratil on the terms of sale within six weeks.
She said: "We want to secure the future of Prestwick Airport and the businesses that depend on it. We want to reassure staff that we will work with them to make the airport a success."
"I can therefore advise Parliament that the Scottish Government has advised the current owners of the airport our intention to commence a process towards acquisition of Prestwick Airport."
Labour MP for Central Ayrshire Brian Donohoe welcomed the move, saying significant investment would be required to develop the airport.
He said: "We need now to ensure concentration is steered towards new routes and new airlines coming into the airport.
"Major money is required to develop the airport which is why I have always said it should be given for a penny to a developer willing to spend the cash."
Private investors have shown interest in Prestwick but none was able to commit on a timescale acceptable to Infratil, said Ms Sturgeon.
Closing the airport would be a "serious and unwelcome development", she told MSPs.
About 300 people depend directly on the airport for employment, while another 1,400 are associated with the facility. A wider "aerospace cluster" at Prestwick supports about 3,200 jobs.
"We believe Prestwick Airport can have a positive future," she said.
"It will require investment and it will take time. However, we believe it can be returned to profitability.
"We also estimate that the cost of closure to the public purse would be very significant and this is an important factor in our decision.
"We are therefore determined that the airport's economic contribution, including the direct and indirect employment opportunities offered by the airport and its related businesses, should be maintained and then enhanced."
Ms Sturgeon stressed that the airport remains open for business.
"This is a point I want to emphasise not just to parliament, but also to the airport staff, to passengers who have already booked to fly, and to those considering Prestwick Airport for their next holiday or business trip," she said.
"Prestwick Airport is, and will continue to be, fully open for business."
The Scottish Government already owns 11 airports, mainly small operations providing links to remote communities.
Marko Bogoievski, chief executive officer at Infratil, said: "In March 2012 Infratil announced its intention to sell its two UK airports, Glasgow Prestwick Airport and Manston Kent Airport, as part of a process to refocus its investment profile.
"Recognising the importance of the airports to their local communities, Infratil's preference has been to secure a new owner with the capacity to support their future success.
"We believe that a Scottish Government acquisition of Glasgow Prestwick Airport achieves that objective and will work proactively with the Scottish Government over the next six weeks towards achieving completion of a transaction."
Bill McIntosh, leader of South Ayrshire Council, welcomed the move.
"Prestwick Airport and the engineering jobs that it supports are hugely important for Ayrshire and for Scotland as a whole," he said.
"We look forward to working closely with the Scottish Government to ensure a long-term future for Prestwick Airport that builds on the substantial contribution it already makes to the Scottish economy."
Adrian Gillespie, managing director of operations at Scottish Enterprise, said: "Prestwick Airport is a vital part of both the local economy in Ayrshire and the wider Scottish economy, supporting a high number of jobs and businesses.
"We are committed to working closely with our partners to achieve a sustainable future for both the airport and all it delivers for our economy and our communities."
Prestwick Airport was bought in 2001 by its current owner, which operated through a subsidiary company, Prestwick Aviation Holdings Limited.
It has faced "many challenges" in recent years, Ms Sturgeon said, with reduced passenger numbers and freight.
But the airport helped bring in about £47.6 million to the economy of Ayrshire and about £61.6 million across Scotland in 2012.
Infratil, which also owns Wellington Airport in New Zealand, reported a total of 121,772 passengers using Prestwick during September, 7% up on the same month last year.
Cargo was especially strong in that month, the firm said, with a total of 1,105 tonnes being shipped, 43% up on September 2012.
Prestwick is Scotland's only airport with a rail connection. It is about 45 minutes away from Glasgow.
Opposition MSPs at Holyrood welcomed the news Prestwick Airport was to be brought into public ownership, but questioned how long it would be before it could be returned to the private sector.
Ms Sturgeon said it could be years before a new, private sector owner for the site was found, adding that owners Infratil had estimated the total annual losses to be in the region of £7 million.
She said it was "clearly an airport that requires a lot of work to turn it around".
Labour infrastructure spokesman James Kelly said closing the airport would have had "drastic implications" for both the local economy and Scotland as a whole.
"From that point of view I welcome the steps the Scottish Government has taken today to seek to take Prestwick Airport into public ownership," he said
But Mr Kelly asked what the impact would be on the Scottish budget, as well as demanding how confident ministers were of finding a buyer in the future.
Meanwhile Tory MSP John Scott, whose Ayr constituency includes the airport, said the Government's action lifted "an immediate threat of closure".
He then pressed the Deputy First Minister on when the airport could be returned to the private sector, as well as asking what investment would be needed in it.
Ms Sturgeon said the Finance Secretary John Swinney would "make such provision as is required within the Scottish Government budget to underpin the commitment we are making to Prestwick Airport".
She added the airport would need Government investment to "put it into the position where at some point in the future it is able to be returned to the private sector".
Ms Sturgeon stated: "Turning Prestwick Airport around bringing it back into profit, will take time, it will take perseverance, it will take patience, it will take investment."
She added: "It is of course our long-term ambition to see Prestwick Airport returned to private sector ownership, but there is a lot of work to be done between now and then in order to ensure that is a viable proposition.
"We are undoubtedly talking a matter of years rather than shorter than that."
Meanwhile Green MSP Patrick Harvie called on the Deputy First Minister to guarantee all staff at the airport would receive the Living Wage, and none would be employed on zero hours contracts
"A publicly owned airport should achieve the same employment standards as the rest of the public sector," he said.
Ms Sturgeon said that ministers had stepped in because of a "desire to safeguard vital employment in Prestwick" and added: "I hope that gives a very strong sense to the importance we attach to these jobs."