Benchmark oil hovered above $89 a barrel while the dollar strengthened against the euro and the yen.
Japan's Nikkei 225 index rose 0.3 percent to 8,787.66. South Korea's Kospi rose 1.1 percent to 1,853.51, benefiting from a decision by the country's central bank to leave its benchmark interest rate unchanged for a third month. Higher interest rates generally drag on stocks by making them a potentially less attractive investment.
Shares in mainland China and Thailand dropped. Hong Kong's Hang Seng fell 0.8 percent to 19,897.15.
Wall Street rose sharply Wednesday after a German court backed the country's role in bailing out smaller, debt-wracked European nations and Italy approved a controversial austerity package to stave off a budget and loan crisis of its own.
The European Central Bank had demanded stiff austerity measures from Italy, but it's not clear whether the package is sufficient. The ECB has spent billions over the last month buying up Italian government bonds to get Italy's borrowing costs lower and help it avoid becoming the next European nation to need an international bailout.
"Relief from events in Europe as Germany's constitutional court blocked an attempt to derail the eurozone rescue package and Italy's passage of austerity measures have boosted risk appetite," Credit Agricole CIB wrote in a research report, adding that the gains appeared to be more a sign of relief than "any major shift in sentiment."
In other Asian markets, benchmarks in Singapore, Taiwan, Indonesia, Malaysia, New Zealand and the Philippines were higher.
Some investors, however, took the jobs data as a sign that Australia's central bank would not hike interest rates in the near future. JB Hi-Fi Ltd., a discount electronics retailer, rose 5.9 percent.
Traders were also looking to Obama's address to the U.S. Congress on Thursday night, where he is expected to announce a $300 billion jobs package made up of tax cuts, state aid and infrastructure spending.
The U.S. job market needs big help. In August, the economy added no new jobs. And the unemployment rate is 9.1 percent, a level more typical for a recession than for a recovery in its third year.
Also awaited was a meeting of the European Central Bank, and whether it would keep its key interest rate at 1.5 percent. Economists speculate the ECB's key rate is now on long-term hold after two increases in April and May, which were aimed at normalizing credit costs after their record lows during the 2007-2009 financial crisis.
Some experts even think that the debt crisis could force the ECB to lower interest rates right back to 1 percent.
Worries over weakness in the U.S. job market and Europe's massive debt predicament have led to recent heavy losses on global stock markets. But investment sentiment improved after a Federal Reserve survey Wednesday showed that the U.S. economy grew modestly in its 12 bank regions in July and August as consumers spent more.
The Dow surged 2.5 percent to close at 11,414.86. The Standard and Poor's 500 index jumped 2.9 percent to 1,198.62. The Nasdaq composite shot up 3 percent to 2,548.94.
Benchmark crude for October delivery was up 9 cents to $89.43 a barrel in electronic trading on the New York Mercantile Exchange. The contract jumped $3.32, or 3.9 percent, to finish at $89.34 per barrel Wednesday.
In currency trading, the dollar rose to 77.37 yen from 77.25 yen in late trading Wednesday in New York. The euro fell to $1.4065 from $1.4093