Ministers said that under the new ownership at least £3 billion will be invested in the green economy over the next three years.
The Government-owned bank, which employs approximately 80 people at its Morrison Street headquarters, was launched in 2012 to channel investment into low carbon development.
But critics have questioned Macquarie Group’s commitment to green projects, with Greenpeace UK labelling the deal ‘a disaster’.
Confirming the £2.3 billion, Climate Change and Industry Minister Nick Hurd said: “The Green Investment Bank has been very successful in attracting private capital to the UK’s green economy.
“It now makes sense to move it into the private sector where it will be free from the constraints of public sector ownership, allowing it to build further on its success.”
The bank has supported almost 100 green infrastructure projects, attracting £3 of third party capital for every £1 invested.
Mr Hurd said the deal was the “best of both worlds”, securing value for the taxpayer, while investment in the green economy will increase.
“GIB has a well-funded new owner that is committed to the bank’s green mission, with a track record of success in green investment and an ambition to grow the business.”
The sale means all taxpayer-funded investment in the bank, including set-up costs, has been returned with a profit, said the Government.
The bank’s brand will be maintained, and the new owners said they will utilise the skills of employees in Edinburgh and London.
The office in Edinburgh will be home to a new green energy business.
David Fass, chief executive of the Macquarie Group said: “The addition of the Green Investment Bank, its people and expertise, strengthens Macquarie’s commitment to the green energy sector.
“Our combined platform will build on the legacy of the Green Investment Bank and, alongside our knowledge of energy and infrastructure, will open further opportunities in low carbon investment both in the UK and further afield.
“We are excited by a business that will take a leading role in the green economy using the specialist knowledge of our teams in Edinburgh and London.”
Former energy secretary Sir Ed Davey, who is seeking a return to Parliament, said the sale was “environmentally irresponsible, and on the eve of the election is politically dubious”.
The £2.3bn sell-off, which specialises in backing environmentally-friendly projects, had faced cross-party opposition from Holyrood.
Bank chairman Lord Smith of Kelvin, said the board had supported the decision to privatise as the best route to securing its long-term future and growing green impact.
He said: “There is a compelling logic in the world’s first green bank joining forces with the world’s largest infrastructure investor.
“When we embarked on this process, we were determined to find a new owner who would build on GIB’s successful history - an owner who would have access to deep pools of capital, a commitment to expand GIB’s activities, and a respect for the unique role GIB has played in the market.
“Macquarie will bring all of this to GIB, along with its own impressive track record of green investments.
“Its vision for the future growth of GIB demonstrates a redoubling of its commitment to a low-carbon economy.”
But Doug Parr, policy director at Greenpeace UK, labelled the sale to the Australian firm “a disaster”.
“At a time when the Government should be shoring up low-carbon industry for post-Brexit Britain, they have given away one of our key tools for advancing green technologies.
“The hole left by the Green Investment Bank will slow our transition to a clean energy system, set us back on reaching our climate targets, and mean more of the jobs from new sectors will go elsewhere.”
Since it was formed, the bank, set up with money from the taxpayer, has invested in projects ranging from offshore wind farms and council LED street lighting to replacing boiler systems in sheltered housing, hydropower schemes, energy from waste plants and energy efficiency investments in distilleries.
The Scottish Government said it will ‘hold the company to account’.
A spokesman said: “The Macquarie Group has pledged to strengthen its commitment to Scotland by providing important new opportunities for the country’s low carbon industries and financial sector. Ministers have pledged to hold the company to account to ensure it delivers on its commitment.”
Economy Secretary Keith Brown said: “I welcome the Macquarie Group’s clear commitment to Scotland and the assurances we have received that it will maintain the Green Investment Bank’s unique identity and its focus on the green projects that are key to Scotland’s continued economic success.“
He added: “The latest statistics show the number of people employed in the renewable and low carbon industries here has risen to 58,500, highlighting the sector’s strong record of success in Scotland.
“This, alongside Edinburgh’s leading role in the financial services sector, means Scotland is uniquely well placed to support investment decision-making in renewables and low carbon projects and I will continue to work with the Macquarie group to underline this.
“The Macquarie Group has indicated that staffing levels at the Green Investment Bank’s Edinburgh offices will be maintained and are likely to increase overall in the longer term. We will follow developments closely in the months and years ahead.”