MAG chief executive Charlie Cornish said the deal is a "significant milestone" and he hoped the Essex site would become a "high quality alternative" to London airports for global travellers.
MAG has been involved in a bidding process with two other candidates after the British Airports Authority (BAA), now Heathrow Airport Holdings, was ordered to sell Stansted on competition grounds. It confirmed its bid to buy Stansted in October.
Late last year, MAG, which is owned by the ten councils of Greater Manchester, agreed a deal to sell 35.5 per cent of its shares to Australian investor Industry Funds Management on the condition it added another airport to the group.
MAG also owns East Midlands airport as well as Bournemouth and Manchester.
Cornish said: "The transaction represents a significant milestone in the achievement of our previously stated strategy of adding a quality airport to the group and delivering long term value to our shareholders.
"MAG has a strong track record of creating shareholder value through strategic investment and enhancing operations to improve customer service and drive increased passenger numbers.
"We will use that expertise at Stansted to ensure the airport can fulfil its potential as a high quality alternative London access point for global air travellers."
The £1.5bn asking price represented an acquisition multiple of 15.6x 2012 EBITDA, MAG said, which compared favourably with similar airport transactions in the UK and reflected Stansted's significant growth potential.
The transaction is expected to reach financial completion by the end of February 2013.
IFM has investments in nine airports in Australia and significant investments in Anglian Water and Arqiva Limited, the UK broadcast and wireless communications infrastructure company.
JP Morgan Cazenove acted as financial adviser and Royal Bank of Scotland as debt adviser to MAG. Gleacher Shacklock acted as financial adviser to IFM.
Bristol-based Burges Salmon advised the Competition Commission on the corporate aspects of the sale after previously advising on the sale of Gatwick Airport in 2009 and Edinburgh Airport in 2012. Burges Salmon did not provide competition law advice, which was handled by the CC.
Heathrow (AH) Ltd's legal advisers were Freshfields Bruckhaus Deringer while Slaughter and May acted for the purchaser.