The news was also welcomed by governments in Japan and China -- two of Europe's biggest debtors -- with them calling for leaders in Greece to act quickly to form a cabinet.
Tokyo stocks jumped 1.77 percent, or 151.70 points, to 8,721.02, Sydney was 1.96 percent higher, or 79.6 points, at 4,136.9, while Seoul climbed 1.81 percent, ending up 33.55 points at 1,891.71.
In the afternoon Hong Kong surged 1.24 percent and Shanghai added 0.58 percent.
Dealers cheered Sunday's result, which saw the New Democracy party win about 30 percent of the vote, just ahead of the radical Syriza group which had campaigned on the promise of tearing up an EU-IMF bailout agreement.
While New Democracy did not win outright it will now be able to form a coalition with the left-leaning, pro-austerity Pasok party.
The news boosted the single currency, which surged to morning highs of $1.2727 and 100.86 yen before paring those gains to sit at $1.2709 and 100.64 yen.
That was up from $1.2644 and 99.47 yen in New York trade late Friday.
The election in Greece was the second in six weeks after May 6 polls failed to produce a government, stoking fears that the political stalemate would paralyse efforts to bring the country back from the brink.
The vote essentially became a referendum on the country's membership of the eurozone, with European leaders warning that a win for Syriza would likely see Athens renege on its debt commitments and ultimately exit the bloc.
Japan welcomed the result but called on Europe to "urgently" strengthen its financial sector and pressed Greece to swiftly form a new government.
"We will be paying close attention to upcoming negotiations to form a coalition," Chief Cabinet Secretary Osamu Fujimura, the government's top spokesman, told reporters in Tokyo.
"Our country hopes that a stable government will be launched early and make progress towards stabilising markets," he added.
China echoed those words, with the official Xinhua news agency saying in a commentary: "The Greek parties must work together to form a new government quickly and convince the Greek voters of the need for painful austerity."
Despite the reprieve offered Sunday, analysts warned that it would not vanquish concerns about the embattled eurozone or Greece's economic future.
Yoshikiyo Shimamine, economist at Dai-ichi Life Research Institute in Tokyo, said the election result meant Greece had just "returned where it was before."
"They are back to the difficult path they had been taking before. I pay respects to the Greek people who chose this road, but it will be a painful, tough road," he told AFP.
"We now have no worries that Greece will leave the euro or declare default suddenly."
Kenichi Hirano, operating officer at Tachibana Securities, told Dow Jones Newswires that the election results were "a net buy incentive for stocks".
However, he added that "Greece's problems are far from over".
And Paul Mackel, head of Asian currency research at HSBC in Hong Kong, said: "There is still some room for disappointment to come in. This is a small bright spot and it could fade briefly."
Athens has been forced to seek bailouts twice in recent years amounting to about 347 billion euros ($442 billion), and European leaders warned Greece it must respect its international debt commitments or risk leaving the euro club.
The EU and IMF pledged support to the next Greek government.
"We are hopeful that the election results will allow a government to be formed quickly," said the presidents of the European Council and the chairman of the European Commission, Herman Van Rompuy and Jose Manuel Barroso.
"We will continue to stand by Greece as a member of the EU family and of the euro area," they added, in a statement issued in the Mexican town of Los Cabos on the eve of the G20 summit of the world's leading economic powers.
A short statement from the International Monetary Fund indicated it will set about a renegotiation of Greece's 130-billion-euro bailout programme.
"We take note of the election results in Greece and stand ready to engage with the new government on the way forward to help Greece achieve its objective of restoring financial stability, economic growth and jobs," it said.
Eyes are now on the G20 meeting in Mexico over the next two days, where Europe's debilitating debt crisis will be discussed ahead of a summit of European leaders next week.
Dealers are also looking to a US Federal Reserve policy meeting this week amid hopes that it will announce fresh monetary easing to kickstart the economy, while details are to be released on Spain's 100-billion-euro bank bailout.
Oil was higher, with New York's main contract, light sweet crude for July delivery advancing 92 cents to $84.95 a barrel and Brent North Sea crude for August delivery rising 94 cents to $98.55.
Gold was worth $1,621.65 an ounce at 0620 GMT, compared with $1,625.50 late Friday.
In other markets:
-- Taipei rose 1.76 percent, or 125.67 points, to 7,281.50.
Taiwan Semiconductor Manufacturing Co surged 3.46 percent at Tw$80.8 while Hon Hai Precision ended up 2.37 percent higher at Tw$86.5.
-- Wellington closed 0.25 percent, or 8.60 points, higher at 3,455.68.
Telecom Corp gained 0.86 percent to NZ$2.49, Fletcher Building was up 0.16 percent at NZ$6.30 and Chorus was down 0.32 percent at NZ$3.10.
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