MSCI's broadest index of Asia-Pacific shares outside Japan added 0.72%.
Chinese stocks were mixed on the day. The Shanghai composite rose fractionally to 2,917.76, while the Shenzhen component declined 0.15% to 9,152.77 and the Shenzhen composite fell 0.125% to 1,548.93.Hong Kong's Hang Seng index gained 0.75%, as of 3:15 p.m. HK/SIN.
The Nikkei 225 rose 0.51% to close at 21,643.53, while the Topix gained 0.47% to finish its trading day in Tokyo at 1,578.63. Shares of game maker Nintendo jumped 4.15% a day after the company announced a cheaper version of its Switch video game console.
In South Korea, the Kospi added 1.06% to close at 2,080.58.
Relations between Tokyo and Seoul remain frosty, with Japan saying last week it would tighten restrictions on exports of three materials used in smartphone displays and chips, over a dispute with Seoul on South Koreans being forced to work for Japanese firms during World War Two.
On Thursday, South Korea announced that up to 300 billion won (approx. $256 million) would be set aside to cope with Japan's export curbs.
Shares of companies potentially affected by the export curbs, such as Samsung Electronics and SK Hynix, rose 1.43% and 3.57% respectively on Thursday.
Australia's rose 0.39% to close at 6,716.10, as the sectors advanced. Banks were under the spotlight, with Australia's corporate regulator threatening on Thursday to prosecute some of the country's largest lenders over their sales of consumer insurance products in the past decade.
Overnight stateside, the S&P 500 breached the 3,000 level for the first time, but pared gains to close 0.5% higher at 2,993.07. The Dow Jones Industrial Average climbed 76.71 points to finish its trading day on Wall Street at 26,860.20, while the Nasdaq Composite saw a record close at 8,202.53.
In testimony to the House Financial Services Committee, Fed Chair Powell said business investments across the U.S. have slowed "notably" recently as uncertainties over the economic outlook linger.
"Many FOMC participants saw that the case for a somewhat more accommodative monetary policy had strengthened," Powell said in prepared remarks. "Since then, based on incoming data and other developments, it appears that uncertainties around trade tensions and concerns about the strength of the global economy continue to weigh on the U.S. economic outlook."
"Anyone fearful that Fed chair Jay Powell might be about to cast aspersions on the likelihood of the FOMC agreeing to cut rates when it next meets on July 30th and 31st needn't have worried," Ray Attrill, head of foreign exchange strategy at National Australia Bank, wrote in a morning note.
"I think it's much clearer that we have a rate cut, one at the end of this month, I think that's pretty baked in now. And then probably followed by one or more rate cuts ... into the next few months," Rajeev De Mello, chief investment officer at Bank of Singapore, told CNBC's "Street Signs" on Thursday.
De Mello added that the Fed was likely to start with a 25 basis point cut.
The U.S. dollar index, which tracks the greenback against a basket of its peers, was at 96.912 after slipping from levels above 97.5 yesterday.
The traded at 108.03 against the dollar after strengthening from levels above 108.5 in the previous session, while the was at $0.6974 after bouncing from levels below $0.695 yesterday.
Oil prices were higher in the afternoon of Asian trading hours, with international benchmark Brent crude futures contract rising 0.45% to $67.31 per barrel, while U.S. crude futures added 0.41% to $60.68 per barrel.
Oil prices surged more than 4% on Wednesday following a larger-than-expected decline in U.S. crude inventories as well as the evacuation of rigs in the Gulf of Mexico by major producers prior to an expected storm.
— Reuters and CNBC's Fred Imbert contributed to this report.