Tokyo's Nikkei index was up 0.56 percent and Sydney's S&P/ASX 200 index rose 0.20 percent, Hong Kong was flat, Shanghai's Composite index was down 0.18 percent and Seoul's KOSPI lost 0.10 percent.
In Japan Honda Motor was among the exporters gaining strongly, as the car firm raised its outlook for US sales this year to 13.1 million vehicles compared with an earlier projection of 12.8 million units.
Kenichi Hirano, operating officer at Tachibana Securities, said gains by US stocks last week as well as an improvement in the euro and dollar against the yen were helping.
In addition worries about rising inflation in some other markets might be giving Tokyo an advantage, he told Dow Jones Newswires.
"Some funds are flowing into Japanese shares out of emerging markets on concerns over tightening measures," said Hirano.
However, worries that China will hike interest rates again to tame strong inflation, possibly early next month, continued to weigh on many markets.
"There might be some technical rebound on the way but we're still likely to be pressured by fears of more rate hikes; that's the main thing that's ongoing right now and has been for the last two trading days," said Gabriel Gan, a senior equities officer at AmFraser in Singapore.
Gains tailed off at the end of last week on Wall Street as worries emerged about personnel changes at Apple and Google, but the Dow Jones Industrial Index closed 0.7 percent higher than the previous Friday's close.
On Asian currency markets the dollar stood at 82.65 yen, marginally up from its 82.60 level in New York on Friday, the euro was at 112.35 yen compared with 112.56, while the euro eased slightly to 1.3592 against the dollar from 1.3611.
On oil markets, New York's main contract, light sweet crude for March delivery, rose 27 cents to $89.38 per barrel. Brent North Sea crude for March was up 28 cents at $97.88.
Analysts said oil traders were looking ahead to this week's meeting of the US Federal Reserve's Open Market Committee on Tuesday and Wednesday for signs of any change in US economic stimulus measures.
Gold opened at $1,347.00-$1,3480.00 an ounce in Hong Kong, unchanged from its Friday closing price.
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