The decision revealed on the SNP website means the area is now in line for a massive £1billion cash injection.
As reported in later editions of the Evening Times yesterday, Mr Cameron has informed council leaders they will get £500m from the UK Government as part of a City Deal project.
That will see the cash spent on infrastructure projects in Glasgow and neighbouring councils to boost the area's economy.
Deputy First Minister Nicola Sturgeon said: "I can confirm the Scottish Government will match in full the funding announced by the UK Government - a proposal which we understand consists of just £15m a year for the first five years, with future years' funding contingent on a review at the end of the first five-year period.
"Of course, the UK's proposed £15m a year is dwarfed by the Scottish Government's on-going investment in Glasgow.
"Investment in the Commonwealth Games, the new Southern General hospital, Fastlink and the Glasgow Subway improve-ments alone amounts to a massive £1.5bn in capital spend - that is Scottish Government investment in jobs and infrastructure happening now.
"We have also provided capital funding of £1.1bn to Glasgow City Council since 2008.
"As well as matching the UK Government funding now, we will also guarantee this funding to Glasgow when Scotland becomes independent."
She added the announce-ment shows the referendum has made the UK Govern-ment sit up and take notice of Scotland.
The Prime Minister said Glasgow is one of the United Kingdom's greatest cities and, like Liverpool, Manchester and Leeds, it not only has a proud history but it also has a fantastic future.
"The city has prospered from industry, innovation and culture but has also been held back by unemployment and skills shortages."
He said that mattered for the whole of the UK, as long-lasting prosperity would only happen if all the country's regions and cities were fulfilling their potential.
The cash announcement by the UK Government has been billed "historic".
It is estimated it will result in 28,000 jobs being created and £3.3bn of private sector investment.
The City Deal project follows lengthy discussions between senior council officials and the Cabinet Office.
There have also been high-level talks with the Treasury, the Department for Business Innovation and Skills and the Depart-ment for Work and Pensions.
Glasgow City Council leader Gordon Matheson said the Prime Minister's announcement was fantastic news.
He said: "It will help us to move to the next level in terms of economic growth for the City Region.
"We have been working closely with the UK Government on this project for some time and I am delighted it has borne fruit.
"The City Deal will bring tens of thousands of jobs, encourage growth and improve our competitive-ness across the whole of the Clyde Valley area for the next 20 years."
South Lanarkshire Council leader Eddie McAvoy said the cash injection offers real hope of a step change in the economic fortunes of the Clyde Valley area.
He added: "This is an historic moment.
THIS means long-term investment in South Lanarkshire, with a kick-start for projects and hopefully thousands more jobs."
Renfrewshire Council believes the new funding will lead to the creation of the long-delayed airport rail link.
Leader Mark Macmillan said: "The City Deal offers the potential for a step change in investment in the west of Scotland and Renfrewshire in particular.
"The UK Government has put a significant financial package in place that will stimulate growth and create thousands of jobs."
Inverclyde leader Stephen McCabe said the investment was partic-ularly welcome as council leaders would decide how and where the cash was spent.
He added: "Inverclyde Council is committed to investing in infrastructure and opportunities to benefit our communities and to provide the right environment to live, work and raise a family.
"The City Deal will provide the opportunity to enhance the economic and social landscape."
North Lanarkshire council leader Jim McCabe, said the City Deal has the potential to create huge long-term benefits.
He added: "We look forward to taking part in further more detailed discussions on these plans in the weeks and months ahead, with the benefits to the people of North Lanarkshire foremost in our minds as decisions are made."
West Dunbartonshire council leader Martin Rooney said: "The offer made by the UK Govern-ment is a fantastic opport-unity for West Dunbarton-shire and neighbouring authorities to grow the economy of this region.
"The chance to bring £500m worth of funding to the area is one I think most people would be excited about."
East Renfrewshire leader Jim Fletcher said: "It will have the dual benefit of providing local investment and infrastructure as well as wider employment opportunities for residents across the wider region."
SEVEN councils will benefit and they are home to more than a third of Scotland's population and 35% of all its jobs.
Under the City Deal, local authorities will have to agree to a phased and proiritised programme of investment. A Commission on Economic Growth will be set up to evaluate the impact of the programme.