Currency traders were also on edge because of uncertainty about talks between European finance ministers over the possible expansion of a bailout fund for troubled eurozone members.
Tokyo added 0.15 percent, or 16.12 points, to end at 10,518.98, Sydney gained 0.81 percent, or 38.7 points, to reach 4,801.8 and Hong Kong rose 0.75 percent by the break.
Shanghai fell 0.33 percent in the afternoon on dealers' continuing fears that China will hike interest rates further to keep a check on inflation.
The regional gains follow a broad sell-off on Monday that came after Beijing hiked the amount of money banks must keep in reserve as the government tries to soak up the liquidity blamed for rising prices.
Eyes are on the release on Thursday of key Chinese economic data, including inflation.
News that Jobs, CEO and figurehead of the iconic Apple brand responsible for the iPod, was to take more time off for medical reasons hit Asian technology firms in morning trade.
Jobs, 55, who underwent surgery for pancreatic cancer in 2004 and a liver transplant in 2009, said he would continue as chief executive "and be involved in major strategic decisions for the company".
But the announcement revived questions about the future of the company.
The news saw Apple fall 6.2 percent on the Frankfurt bourse, while the impact on the Dow will not be known until later Tuesday as Wall Street was closed Monday for a holiday.
In Tokyo, Softbank, the sole network provider for the iPhone in Japan, fell early on, but recovered to end up 0.75 percent, while Tokyo Electron also fell before ending up 0.17 percent. Kyocera ended down 0.7 percent.
In Taipei Hon Hai, the parent of Foxconn that assembles several Apple products, was flat after shedding almost one percent in the morning.
But rivals gained, with Sony up 0.48 percent.
But Kenichi Hirano, operating officer at Tachibana Securities, warned: "This may trigger a broad profit-taking in the US equity market."
On forex markets the euro fell amid concerns over the two-day European finance ministers' meeting that started Monday with divisions among members.
The single currency clawed back earlier losses to hit $1.3315 in Tokyo afternoon trade, from $1.3292 in London late Monday, while it was flat at 109.90 yen.
The dollar fetched 82.54 yen versus 82.68 yen in late London trading.
Ministers are discussing increasing the European Financial Stability Facility, an emergency lending mechanism for fiscally troubled eurozone countries that was created last year in the wake of the Greek crisis.
Analysts have warned the fund would be too small if bigger economies such as Spain, Italy or Belgium need help amid fears Portugal could be next to tumble into the financial abyss, causing a domino effect.
"If they don't release anything new, the risk to the euro will be to the downside if anything, as investors are still concerned about the European debt problems," Minoru Shioiri, chief manager of forex trading at Mitsubishi UFJ Morgan Stanley Securities, told Dow Jones Newswires.
On oil markets New York's main contract, light sweet crude for February delivery, dipped 47 cents to $91.07 per barrel and Brent North Sea crude for March was up 30 cents at $97.73.
Gold opened at $1,361.60-$1,362.60 an ounce in Hong Kong, marginally down from Monday's finish of $1,362.00-$1,363.00.
In other markets:
-- Seoul edged down 0.16 percent, or 3.37 points, to 2,096.48.
-- Taipei rose 62.91 points, or 0.70 percent, to 8,988.00.
-- Manila 1.82 percent, or 75.57 points, to 4,072.59.
Property developer Megaworld was off 6.4 percent at 2.20 pesos, Metropolitan Bank fell 2.7 percent to 63 and Manila Electric shed 5.0 percent 266.
-- Wellington closed 0.31 percent, or 10.56 points, lower at 3,343.74.
Fletcher Building shed 0.5 percent to NZ$7.86 and Air New Zealand fell 3.3 percent to NZ$1.45. But Telecom added 0.4 percent to NZ$2.27.