Under the three-year aid plan, announced after 11 hours of talks in Brussels, the European Commission will make about $77bn available, while countries from the 16-nation euro zone promised loans and loan guarantees of $570 billion.
The International Monetary Fund (IMF) is to match at least half of the EU's total contribution, or about $324bn.
Greece's borrowing costs fell sharply after the deal and stocks rallied.
The difference between yields on Greek 10-year bonds and their benchmark German equivalents was at 5.21 percentage points, down massively from a record 10.25 points on Friday.
The general share index on the Athens stock exchange gained 9.68 per cent in midday trading, reaching 1,788.32 points.
"It was a rescue package of huge dimension. Nobody expected this amount of money to be put on the table," Vassilus Vlastarakis, the head of research at Beta securities, said.
"Of course the market reacted very positively. It started around eight-nine percent in the plus side. And we expect it to last."
The EU's slow response to the crisis and its failure to keep Greece from reaching the brink of bankruptcy triggered the slides in the euro and global stocks last week, and intensified fears the crisis would spread.
The euro had sunk to a 14-month low against the dollar last week, but on Monday it recovered to $1.3.
Al Jazeera's Jonah Hull, reporting from the London Stock Exchange, said that investors had welcomed the stability package.
"It is being called a relief rally here. A response to the deal concluded in Europe and a reflection of the importance of the euro zone to business prospects," he said.
But Hull said that with the UK, which is not in the euro zone, still undergoing a political deadlock after inconclusive elections last week the rally may quickly relapse there.
He quoted one senior broker as saying: "The UK is on thin ice and we can hear it creaking."
The two main opposition political parties in the UK are currently wrangling over the terms for forming a possible coalition government.
UK markets shed $50bn on Friday due to concerns over stability in the euro zone and the political situation at home.
European central banks began buying government bonds from those countries with most uncertainty on Monday as their part of the agreement.
The German and Finnish central banks confirmed that they had started buying the bonds, with all euro zone central banks to follow suit.
The safety net was meant to fend off dealers on the financial markets from speculating on countries with bloated budgets, such as Portugal, Spain and Ireland.
Greece has committed to significant austerity measures - tax increases and spending cuts - to receive separate emergency loans of $140bn from the EU and IMF to solve its problems.
George Papaconstantinou, Greece's finance minister, said on Monday that his country would now be able to pay off creditors.
"Over the next few days, payment will start of the first section of the loans from the European Union and the International Monetary Fund, so that the country has no problems whatsoever with its borrowing needs and servicing its debt this month, and for months to come," he said.
Barnaby Phillips, Al Jazeera's correspondent in the Greek capital Athens, said once Athens starts to receive the funds there would be little chance that it could meet public calls back down on the austerity measures.
"It is almost inconceivable in fact," he said.
"The result is that Greece is probably doomed to deep recession that may last years and it is doomed to profound social changes and indeed social tensions.
"The up side, if you like, is that assuming that Greece is eligable for all these new funds that are being pledged on top of the $140bn Greece was already granted last week it is harder to see how Greece will default on its debt further down the road."
However, Phillips said that the Greek public remained acutely uncertain about its future.
"We will have to see in the weeks and months ahead which way public opinion goes. There are no big protests planned in Athens today but we are hearing there will be more later in the week."