The cash from the Heritage Lottery Fund (HLF) means work can begin, subject to planning permission, on the £10million project at the start of next year.
The HLF funding means a further £4million will be released by the Scottish Government, Scottish Borders Council, Scottish Enterprise, Historic Scotland and a number of private individuals.
Chairman of the Abbotsford Trust, Lord Sanderson, said: "Our plans for Abbotsford aim to create a legacy for one of Scotland's most important sons.
"We are obviously delighted with today's announcement which means that we have now raised £9million of the estimated £10million we believe is required to make our vision for Abbotsford a reality."
The former Scottish Conservative Party chairman added: "This announcement means that we can start to put our plans into action and create a world-class visitor experience that will make the Borders, Scotland and the UK extremely proud, and which we believe will deliver lasting economic and cultural benefit."
Among the plans are repair and restoration of the 19th-century building's roof, described as "vital" by the project's design leader Mark Hopton.
They also hope to create six-bedroom self-catering accommodation in the Hope Scott Wing, new intrepretation facilities, a learning suite for schools and a new car park.
However, the most controversial section of the renovation plans is the new visitor reception, which drew comments such as "a really ugly 1960s building" during a public consultation at the Wynd Theatre, Melrose, in March.
But Scottish Government culture minister Fiona Hyslop dismissed the reception concerns.
She said: "When I visited Abbotsford earlier this year I was particularly impressed by how the new facilities at the visitor centre will not just benefit people going to the house but will offer a base for wider tourism across the Borders, linking walking trails and showcasing what the region has to offer."
March's public consultation heard that Abbotsford is currently losing a significant amount of money attributed to the way the house is set up.
Visitor figures have fallen to 24,000 per year from around 80,000 in the late 1980s.
But following the improvements, trustees hope Abbotsford will attract 40,000 tourists per annum.
Head of the HLF in Scotland, Colin McLean, said his organisation had been working with Abbotsford for 12 months on the plans.
He added: "Sir Walter Scott is recognised as the man who re-invented our Scottish national identity then presented it to the world through his literature.
"Through this project, Abbotsford can once again take its place on the world's stage as a tourist destination and valuable education resource.
"Today we can celebrate that this exceptional piece of Scottish history is going to have a vibrant and sustainable future, bringing income to the Borders and to Scotland."
Meanwhile, Michael Moore believes the Borders economy will benefit from the restoration of Scott's home, with his name helping to raise £10million nationally during the Year of Homecoming in 2009.
The MP for Tweeddale, Ettrick and Lauderdale said: "The announcement of a £4.8million grant by the Heritage Lottery Fund is great news as it means the restoration of one of Scotland's iconic and well-known houses can now take place. It also means that funding from the Scottish Borders Council and the Scottish Government will also be released to help secure the future of the house for generations to come.
"Abbotsford House was the home of Sir Walter Scott who is regarded as one of the most influential writers and poets of his time. He was the first English language writer to have an international career during his own lifetime and it is only right that we retain a link with the man and his work.
"The restoration of the house will lead to another tourist attraction for the Borders which will help the local economy once completed."
Around £1million still needs to be raised to meet the £10million target for the project, but is expected to be met by donors who were awaiting news on Abbotsford's HLF funding bid.
And trustees are hoping to raise a further £3million as an endowment to safeguard the house's future.Lord Sanderson, chairman of the Abbotsford Trust, and Colin McLean, head of the Heritage Lottery Fund in Scotland, at Abbotsford for the grant announcement. Also pictured are: Andrea Longson, Faculty of Advocates; David Parker, Scottish Borders Council; Lucy Casot, HLF; Jacquie Wright, executive manager, Abbotsford; George Dalgleish, trustee; Cllr Graham Garvie; Tony Taylor, trustee; Jason Dyer, fundraising director, Abbotsford; Mark Hopton, LDN Architects; Andrew Douglas-Home, trustee; Paul Jardine, Jura Consultants.