Mainland Chinese stocks were higher on the day, with the Shanghai composite up 0.76% to 3,382.32 while the Shenzhen component advanced 0.956% to 14,015.02. Hong Kong's Hang Seng index gained about 0.7%, as of its final hour of trading.
Shares in Australia edged higher by their close, with the S&P/ASX 200 gaining 0.66% to 6,643.10.
MSCI's broadest index of Asia-Pacific shares outside Japan rose 0.72%.
In corporate developments, shares of South Korea's LG Electronics surged 29.61% after the firm announced Wednesday a joint venture agreement with Canadian automotive supplier, Magna International, to manufacture electric vehicle components.
Trump on Tuesday said the $900 billion Covid relief bill passed by U.S. Congress was an unsuitable "disgrace."
While the U.S. president did not threaten to veto the legislation, he asked to be sent a "suitable bill or else the next administration will have to deliver a Covid relief package." The latest development could delay the deployment of funds to Americans who are struggling amid the coronavirus pandemic.
A new Covid strain in the U.K. has also been weighing on investor sentiment globally in recent days. It comes at a time when vaccines are starting to be rolled out and major economies around the world have seen a resurgence in daily coronavirus infections.
In the Bank of Japan's October monetary policy meeting minutes released Wednesday, one member expressed concern that "deflation might take hold" if Covid-19 spreads again and pushes down economic activity, so this "possibly warranted attention" in guiding monetary policy.
Overnight on Wall Street, the S&P 500 saw a third straight session of losses as it declined 0.2% to 3,687.26. The Dow Jones Industrial Average slipped 200.94 points to close at 30,015.51. The Nasdaq Composite outperformed as it rose 0.5% to close at 12,807.92, a new record.
The U.S. dollar index, which tracks the greenback against a basket of its peers, was last at 90.445 following an earlier high of 90.666.
The Japanese yen traded at 103.40 per dollar, still stronger than levels above 104 seen last week. The Australian dollar changed hands at $0.7565, in a trading week that has seen it at levels around $0.76 to below $0.75.
— CNBC's Mike Calia and Maggie Fitzgerald contributed to this report.