Tokyo jumped 1.08%, or 164.10 points, to finish at 15,326.20 and Shanghai edged up 0.10%, or 2.05 points, to 2,050.38.
Sydney dipped 0.37%, or 19.85 points, to 5,375.9 and Seoul fell 0.16%, or 3.21 points, to 1,999.00.
Hong Kong and Bangkok were closed for public holidays.
The Bank of Japan said its quarterly Tankan survey of business confidence had edged down in April-June — the first dip in 18 months — after consumer spending was hit by a sales tax hike.
The closely watched report, a reading of large manufacturers, fell to plus 12 from plus 17 in the previous three months. It also marks the first decline since Prime Minister Shinzo Abe took office in late 2012.
However, while the study showed weakening business sentiment, traders took heart from data showing firms raised their combined investment plans for the financial year ending March to a 7.4% increase, from a previous 0.1% rise forecast in the March survey.
"While today's Tankan points to a contraction in GDP of around 0.5%... last quarter, the survey suggests that the recovery will resume in the second half of the year," said Marcel Thieliant of Capital Economics.
In China, the official purchasing managers index of manufacturing activity in June came in at 51, slightly higher than May's 50.8, adding to hopes a slowdown in the world's number two economy has bottomed out.
A figure above 50 points to growth while anything below suggests contraction.
A separate PMI by banking giant HSBC hit 50.7, up from 49.4 in May and the first time it has been in positive territory this year, thanks to a series of mini-stimulus measures by Beijing.
"The economy continues to show more signs of recovery, and this momentum will likely continue over the next few months, supported by stronger infrastructure investments," HSBC said. However it warned "there are still downside risks from a slowdown in the property market".
In the United States, news that pending home sales in May saw their highest month-on-month gain in four years was offset by data showing a slowdown in economic activity in the Chicago area.
The Dow slipped 0.15%, the S&P 500 edged down 0.04% and the Nasdaq added 0.23%.
On currency markets, the dollar and euro edged up against the yen in line with a rise on Japan's Nikkei owing to growing confidence in higher yielding, riskier assets.
The dollar bought 101.47 yen in afternoon Tokyo trade compared with 101.31 yen in New York Monday afternoon.
The euro was changing hands at $1.3687 and 138.90 yen against $1.3694 and 138.73 yen.
Oil prices edged higher. US benchmark West Texas Intermediate for August delivery rose 36 cents to $105.73 while Brent crude gained 14 cents to $112.50.
Gold fetched $1,325.00 an ounce at 1055 GMT compared with $1,312.17 late Monday.
In other markets, Jakarta closed up 0.13%, or 6.24 points, at 4,884.83. Miner Aneka Tambang gained 0.92% at 1,100 rupiah, while Bank Permata fell 0.38% at 1,300 rupiah.
Manila eased 0.26%, or 17.70 points, to close at 6,826.61. Philippine Long Distance Telephone fell 0.13% to 2,984 pesos and Ayala Corp closed 0.39 percent lower at 645 pesos.
Mumbai rose 0.40% or 102.57 points to end at 25,516.35 points. Bajaj Holdings rose 9.24% to 1,380.85 rupees while Gujarat State Petronet rose 9.01% to 98.00 rupees.
Singapore fell 0.40%, or 13.03 points, to close at 3,242.64. Property developer CapitaLand tumbled 1.25% to S$3.16 and banking giant DBS rose 0.36% to S416.81. Taipei rose 0.52%, or 48.85 points, to 9,441.92.
Taiwan Semiconductor Manufacturing Co added 1.19% to T$128.0 while Hon Hai Precision was 2% higher at T$102.0.
Wellington was slightly higher, adding 4.79 points to 5,146.26.