The Office for National Statistics said there had been a "fairly strong" performance across all sectors.
The data builds on a 0.7% GDP rise in the April-June period and is the best quarterly performance since 2010.
Chancellor of the Exchequer George Osborne tweeted: "This shows that Britain's hard work is paying off & the country is on the path to prosperity."
Deputy Prime Minister Nick Clegg said the figures "show that we are firmly on the road to economic recovery".
The ONS data for construction was up 2.5% over the quarter, the second successive quarter of growth after a volatile performance over the past year.
The BBC's chief economics correspondent, Hugh Pym, said: "This could signal that a recovery in that sector is really under way."
George Osborne: "Britain's hard work is paying off"
House-builders have been buoyed by the Government's Help to Buy scheme, which recently launched a new phase offering mortgage guarantees.
The ONS said that production grew by 0.5%, though this remains 12.8% off its 2008 level, while within this, manufacturing improved 0.9% in the third quarter.
The services sector, which represents three-quarters of economic output, grew by 0.7%. Output from services is now 0.4% above its pre-crisis peak in the first quarter of 2008.
'Path to prosperity'
Friday's figure is the first estimate, and could be revised up or down in subsequent months.
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Analysis
The fastest growth in three years - it's what the Treasury has been hoping for after ringing the date of this release in the diary months ago.
That means three successive quarters of expansion and what looks like a solid recovery. All three main sectors - services, manufacturing and construction -grew in the third quarter.
Construction, after proving a major drag on the economy last year, was up 2.5%, though data in this notoriously volatile sector is always subject to revision.
But overall economic activity is still not back where it was before the recession. The important services sector is higher than the previous peak, but manufacturing and construction some way below. Further evidence is needed before this can be judged a truly balanced recovery.
Mr Osborne told the BBC: "It shows that we are on the path to prosperity. Lots of risks remain, so we have to stick with the economic plan that has got us this far.
"What's encouraging about these figures is that it's not just services that are growing. Construction is growing and manufacturing is growing."
Shadow chancellor Ed Balls said: "After three damaging years of flatlining, it's both welcome and long overdue that our economy is growing again.
"But for millions of people across the country still seeing prices rising faster than their wages, this is no recovery at all."
Despite the further evidence that recovery is gaining hold, some economists remain cautious.
The Institute of Directors' chief economist Graeme Leach said: "The outlook looks better than at any time since the onset of the financial crisis. Indeed, our members have more confidence in the economy than at any time since 2008.
"However, strong headwinds remain and the annual growth rate year on year is nothing to get too excited about yet. Though inflationary pressures are likely to remain benign, debt and inflation are rising faster than earnings.
'More to be done'
Ed Balls welcomed the GDP figures but said the government should do more to improve living standards
"By far the biggest challenges remain on the supply side, not the demand side. Supply side constraints mean that the current growth spurt is unlikely to extend beyond next year.
"This stage of our economic recovery is likely to be short and sweet, instead of long and strong."
The director-general of the British Chambers of Commerce, John Longworth, said: "This is the highest quarterly increase we've seen in three years, so the economy is clearly moving in the right direction.
"But we are still behind a number of advanced economies, such as the US and Germany, that have managed to recover the output lost during the economic downturn.
"Much more needs to be done to transform our economy from being good to being really great."