Japan's Nikkei stock index added 0.7%, getting a tailwind from a weaker yen and better-than-expected April exports.
Asian shares got off to a slow start on Monday after U.S. Federal Reserve Chair Janet Yellen indicated that the central bank was poised to raise interest rates this year if the economy keeps improving as expected.
Activity was likely to be thin, with the UK, U.S. and many European markets shut for holidays.
MSCI's broadest index of Asia-Pacific shares outside Japan was up 0.1 percent in early trade, with weakness in Southeast Asia offset by rises in China, Australia and South Korea.
Japan's Nikkei stock index added 0.7%, getting a tailwind from a weaker yen and better-than-expected April exports.
U.S. shares fell and Treasury yields and the dollar rose on Friday, after the U.S. Labor Department's gauge on core consumer goods prices rose by 0.3 percent last month, bringing the year-on-year rise to 1.8%, the highest since October.
"For the first time in nearly two months, investors began rewarding the dollar for good economic data rather than punishing it for weaker data," Marc Chandler, global head of currency strategy at Brown Brothers Harriman in New York, said in a note to clients.
In a speech to a business group, Yellen said she expected economic data to strengthen and noted that some of the U.S. economy's weakness at the start of the year might be due to "statistical noise."
Greece cannot make debt repayments to the International Monetary Fund next month unless it manages to reach a deal with its lenders, its interior minister said on Sunday, in the most explicit remarks so far from Athens about the likelihood of default if talks fail.
That kept pressure on the euro, which was down about 0.2% at $1.0996, pushing to its lowest levels since late April.
The dollar was slightly higher against its Japanese counterpart at 121.63 yen, trading at its highest levels since mid-March.
Oil futures steadied after skidding ahead of the long U.S. holiday weekend, giving up about 2 percent on Friday as a rallying dollar and profit-taking took their toll.
U.S. crude added about 0.2% to $59.83 after eking out a small weekly rise to extend its weekly gains for a 10th straight week.
Brent was down about 0.1% at $65.31 after dropping 2.1% for the week.
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