The rally followed sharp rises in European markets, which came after France and Germany proposed a fiscal union ahead of a summit on 9 December.
A union would mean binding limits on eurozone government borrowing.
Markets hope that the agreement will open the way for the European Central Bank to bail out Italy and Spain.
On Wall Street, the Dow Jones index rose 2.6% to 11,523.01 points, while the broader S&P 500 index rose 2.9% to 1,192.57 points.
US markets were also helped by data suggesting that recent consumer spending has been strong.
The US National Retail Federation said sales during the Thanksgiving holiday weekend climbed 16% to a record $52.4bn.
In Europe, the FTSE closed up 2.8%, Germany's Dax was 4.6% higher and France's Cac added 5.4%, led by banks and other financial stocks.
These gains followed strong rises in Asian markets overnight.
The stock market rally came despite ratings agency Moody's warning that the eurozone crisis threatens the credit rating of all the bloc's members.
Moody's said the eurozone could still face a number of shocks as countries struggled with high debt levels, denying them access to markets and requiring further bailouts.
"This would very likely cause those countries' ratings to be moved into speculative [junk] grade in view of the solvency tests that would likely be required and the burden-sharing that might be imposed if, as is likely, support were to be needed for a sustained period," Moody's said in its report.
On plans for fiscal union, the German Finance Minister, Wolfgang Schaeuble, said his government was working closely with the French on the proposal.
"We are working intensively for the creation of a stability union," he said in a statement.
"That is what we want to secure through treaty changes, in which we propose that the budgets of member states must observe debt limits."
The minister acknowledged on Sunday that it may not be possible to get necessary treaty changes agreed by all 27 European Union members states.
In the UK, Conservative backbench MPs in the coalition government have been demanding a referendum on the UK's EU membership as a condition of ratifying any new treaty, while Prime Minister David Cameron has said he will use it as an opportunity to demand new policy opt-outs from Brussels.
There has been speculation that the 17 eurozone governments may go it alone - passing separate bilateral agreements on the new fiscal rules, thereby avoiding the need for EU treaty changes, and circumventing a UK veto.
The unprecedented move comes as many eurozone governments, including Italy and Spain, have faced sharply rising borrowing costs in nervous financial markets.
"We are in an economic war with a number of powerful speculators who have decided that the end of the euro is in their interest," claimed the French Agriculture Minister Brun Le Maire on Monday.
By Robert PestonBusiness editor, BBC News
The price for the Italian government of borrowing remains prohibitively high. Which implies that even if the Italian government isn't talking to the IMF about financial support, it may only be a matter of time.
Are markets betting on IMF bailout of Italy?
Italy sold 567m euros of 12-year bonds at an auction on Monday, with borrowing rates rising to 7.2%, compared with the 4.5% rate at the last equivalent auction.
On Tuesday, Italy will auction 8bn euros of debt, including 10-year-bonds.
However, the country's implied 10-year cost of borrowing, based on bond market prices, fell modestly on Monday, from 7.26% to 7.14% by mid-afternoon, in line with the positive sentiment on stock markets.
However, a level above 7% is widely considered unsustainable for Italy in the long-run.
Also on Monday, Belgium sold 2bn euros of government debt at auction, paying a yield of 5.7% on its 10-year bonds compared with 4.37% at an auction in October.
Last Tuesday, Belgium saw its credit rating downgraded by ratings agency Standard and Poor's.
Bond auctions are watched carefully by investors for signs that markets may be losing confidence in the ability of governments to fund their borrowing.
Over the weekend, the German newspaper Die Welt reported that a subset of eurozone countries with the highest credit ratings, including France and Germany, may seek to issue debt jointly.
However, the plans were denied by the German government on Monday.
"There are no plans for 'Triple-A' bonds or 'Elite bonds' as the article says," said the German finance ministry.
On Monday, President Barack Obama warned that failure of the eurozone to tackle the crisis could damage the US economy.
Speaking after meeting European Union officials, he said: "If Europe is contracting, or if Europe is having difficulties, then it's much more difficult for us to create good jobs here at home."
Meanwhile, the International Monetary Fund (IMF) denied claims that it was in bailout talks with Italy's government.
Italy's La Stampa newspaper had said over the weekend that the IMF was preparing a 600bn euros ($794bn; Â£515bn) loan.
However, an IMF spokesman said: "There are no discussions with the Italian authorities on a programme for IMF financing."
Italy has come under intense scrutiny from the markets in recent weeks. The fear is that, given its enormous debt levels and slow rate of growth, the country may find it tough to pay back its debt and to raise further cash.
An IMF team is expected in Rome shortly to monitor the progress of the new government in cutting the deficit.
The technocratic government of Prime Minister Mario Monti is expected to announce new measures early next month.
The Italian government's debt stands at 118% of the gross domestic product (GDP).
To make matters worse, its annual economic growth rate over the past 15 years has been just 0.75%, and the country is now forecast by the Organisation for Economic Co-operation and Development to be entering recession.
Wall Street rebounded from seven days of losses on hopes that eurozone leaders will stitch together a new deal to help the bloc's ailing economies.