Oil gained to stay above $96 a barrel as concern among traders that the U.S. Federal Reserve might speed up the withdrawal of its economic stimulus diminished.
Tokyo's Nikkei 225, the regional heavyweight, soared 3.8 percent to 13,363.18 while Hong Kong's Hang Seng Index fell 0.4 percent to 21,509.91. Markets in China and Australia were closed for holidays.
Traders were encouraged after a closely watched U.S. Labor Department measure of employment showed the world's largest economy added 175,000 jobs in May, better than the expected 165,000 increase.
"The relief from U.S. jobs was palpable," said Mizuho Corporate Bank in a report. It said gains were " gentle enough to avert concerns" the Fed might be prompted to withdraw its "quantitative easing."
The Fed is trying to encourage corporate investment and consumer spending by purchasing $85 billion a month in government bonds to push down market interest rates.
Markets have been uneasy over the past two weeks as traders scrutinized comments by Fed officials for hints about when that will end. That would nudge interest rates higher.
"There has been a bit of a return to trading on expectations maybe the tapering off is not going to happen as quickly," said Herald van der Linde, HSBC's head of Asian equity strategy.
The Labor Department's monthly employment survey is one of the most important economic gauges and is closely watched by investors. It can cause big moves in financial markets, especially if the report shows that employment is stronger or weaker than economists were expecting.
Also Monday, South Korea's Kospi gained 0.3 percent to 1,928.89 while Taipei's Taiex added 0.8 percent to 8,162.89. Markets in Singapore, Manila and New Zealand also rose. Indonesia's benchmark fell.
U.S. and European markets gained Friday on similar expectations for QE to continue. The Dow Jones Industrial average had its best day in five months, gaining 1.4 percent.
Monday's gains came despite China's weekend data that showed trade, retail sales and other activity in May weaker than expected, fueling concerns about the country's shaky economic recovery.
"This puts some question marks over the strength of the recovery in China," said van der Linde.
In Japan, Prime Minister Shinzo Abe told broadcaster NHK on Sunday he plans to roll out new corporate tax cuts later this year.
That came on top of the Japanese government's upward revision in last quarter's growth estimate to 4.1 percent from an earlier 3.5 percent.
In currency markets, the euro fell to $1.3196 from $1.3221 late Friday in New York. The dollar edged up to 98.23 yen from 97.43 yen.
Benchmark oil for July delivery rose 16 cents to $96.19 per barrel in electronic trading on the New York Mercantile Exchange, up from the previous close of $96.03. The contract added $1.27 on Friday.