Oil prices slipped after rising on Monday in response to a vote for independence from Ukraine in the country's east that has fuelled fears of a civil war.
Tokyo jumped 1.95%, or 275.92 points, to 14,425.44, Sydney rose 0.91%, or 49.8 points, to close at 5,498.2 and Seoul gained 0.92%, or 17.99 points, to 1,982.93.
Hong Kong added 0.41%, or 90.77 points, to 22,352.38 and in late trade Mumbai's Sensex was up 1.55%.
But Shanghai slipped 0.10%, or 2.14 points, to 2,050.73 -- giving up early losses, as data from Beijing showed a slowdown in growth in industrial output, fixed asset investment and retail sales.
In New York the Dow and S&P 500 hit all-time highs and the Nasdaq enjoyed a boost as dealers raced back into technology shares after a recent sell-off that has seen some household names such as Twitter and Netflix tumble.
Analysts said there was no clear reason for the buying but suggested there was a sense that some stocks had been oversold as investors worried the sector was overvalued.
The Dow rose 0.68%, the S&P 500 gained 0.97% and the Nasdaq added 1.77%.
The upbeat sentiment filtered through to currency markets, where investors sought out higher yielding, riskier assets, pushing the yen -- considered a safe bet -- lower.
In afternoon trade the dollar bought 102.30 yen, compared with 102.09 yen in New York Monday.
The euro bought $1.3762 and 140.81 yen, compared with $1.3758 and 140.50 yen.
The single currency has been under pressure since last week when European Central Bank chief Mario Draghi said he was ready to ease monetary policy soon as the currency bloc suffers from continuing weak inflation.
Analysts said the comments would likely force the ECB to act at next month's policy meeting or risk losing its credibility with the markets.
In India exit polls Monday indicated Modi's right-wing opposition alliance had secured votes to form a government without needing new coalition partners.
Investors are hoping Modi will introduce more pro-business policies in a bid to jumpstart the Indian economy, which is expanding at its slowest rate in a decade.
"The expectation is that (the BJP alliance) will get to form the government comfortably and even if they need more allies they will not present a stumbling block for reforms," Harendra Kumar, head of Mumbai-based brokerage Elara capital, told AFP.
On oil markets prices were lower following Monday's gains, after pro-Russian separatists claimed a victory in weekend referendums in Donetsk and Lugansk.
Kiev denounced the balloting as a "criminal farce", while the United States and European Union called the votes illegal.
However, the elections have fanned fears of a violent breakup of Ukraine and the possibility of a civil war on Europe's eastern edge.
Ukraine is a major conduit for Russian oil and gas exports to Europe, and any escalation of the conflict could disrupt supplies and send prices soaring, analysts say.
The US benchmark, West Texas Intermediate for delivery in June, shed 10 cents to $100.49 a barrel, while Brent North Sea crude for June fell 25 cents to $108.16 a barrel.
Gold fetched $1,291.30 an ounce at 0810 GMT compared with $1,294.30 late Monday.
In other markets:
-- Taipei rose 0.11%, or 9.33 points, to 8,817.94.
Taiwan Semiconductor Manufacturing Co closed 1.69% higher at Tw$120.5 while Hon Hai Precision gained 0.23% to Tw$87.9.
-- Wellington was up 0.72%, or 36.92 points, at 5,199.34.
Telecom added 1.70% at NZ$2.68 and Fletcher Building rose 0.75 percent to NZ$9.39.
-- Manila added 0.61%, or 41.47 points, to 6,852.81.
Puregold Price Club rose 4.35% to 48 pesos while Ayala Land added 0.79% to 32 pesos.
-- Bangkok, Singapore and Kuala Lumpur were closed for public holidays.