Japan's Nikkei 225 index surged 2.6 percent to 14,472.90 and Australia's S&P/ASX 200 was up 1.5 percent to 4,881.70.
The move higher came a day after concerns over a sustained China slowdown prompted a selloff in Asia.
Other regional markets rose only moderately, reflecting continued caution. South Korea's Kospi inched up 0.7 percent to 1,830.35 and Taiwan's Taiex rose 1.1 percent to 7,971.18. Hong Kong's Hang Seng added 0.5 percent to 20,683.01.
Mainland Chinese shares made small gains, with the Shanghai Composite up 0.4 percent to 1,965.45. The smaller Shenzhen Composite Index rose 0.4 percent to 892.93.
"Investors now worry about the economic outlook with no signs showing a recovery," said Li Jianfeng, an analyst at Caida Securities in Shanghai.
European markets extended the previous days's gains. Britain's FTSE 100 was up 1 percent to 6,515.02 and France's CAC 40 gained 0.7 percent to 3,851.17. The DAX in Germany was up 1 percent to 8,051.03.
Wall Street has been buoyed by stronger-than-expected U.S. hiring figures released Friday that suggested the world's biggest economy is on a firm footing. The strong jobs growth has outweighed jitters about the prospect that the Fed will soon start winding down its unprecedented support for the economy
Futures augured more gains on Wall Street. Dow futures were up 0.4 percent to 15,218 and S&P 500 futures rose 0.5 percent to 1,643.
Given the prevailing focus on the U.S., the key day this week will likely be Wednesday, when the minutes of the last policy meeting of the Fed are published. The Fed's chairman, Ben Bernanke, is also due to deliver a speech.
Benchmark crude for August delivery was down 3 cents to $103.11 a barrel in electronic trading on the New York Mercantile Exchange.
Among major currencies, the euro rose to $1.2872 from $1.2866 late Monday. The dollar rose to 101.24 yen from 100.90 yen.
___
Researcher Fu Ting in Shanghai contributed.
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