Tokyo advanced more than 7% and Frankfurt opened 6% higher. London, Shanghai, Paris and Hong Kong also rose.
On Wall Street, futures for the benchmark S&P 500 index and Dow Jones Industrial Average gained 5%.
Traders were encouraged by the Fed's promise to buy as many Treasurys and other assets as needed to keep financial markets functioning.
On Monday, Wall Street fell 3% after Congress failed to approve an economic support package. It would send checks to U.S. households and offer support for small businesses and the hard-hit travel industry, but Democrats say it favors companies too heavily at the expense of workers and public health.
"Asian investors like what they see from an all-in Fed which is being viewed in a very impressive light for both Main and Wall Street even as the U.S. congress dithers," said Stephen Innes of AxiCorp. in a report.
In early trading, the FTSE 100 in London rose 3.6% to 5,173.77 and Frankfurt's DAX rose 6% to 9,278.51. The CAC 40 in Paris added 4.4% to 4,088.00.
In Asia, Tokyo's Nikkei 225 rose 7.1% to 18,092.35 and the Kospi in Seoul surged 8.6% to 1,609.97. The Shanghai Composite Index was 2.3% higher at 2,722.44 and Hong Kong's Hang Seng gained 4.5% to 22,663.49.
Australia's S&P-ASX 200 gained 4.2% to 4,735.70 and India's Sensex added 3.1% to 26,776.71. New Zealand rose 7.2% and Singapore added 5.3%.
Also Tuesday, a measure of Japanese factory activity fell to its lowest level since 2009 in another sign of the punch to manufacturers from the coronavirus. The preliminary version of this month's Jibun Manufacturing Purchasing Managers' Index fell to 44.8 from February's 47.8 on a 100-point scale on which numbers below 50 show activity contracting.
The Fed's promise goes beyond the $700 billion in asset purchases announced last week.
The central bank said it will buy a wide range of investments, including corporate bonds for the first time, to improve trading in markets that help home buyers finance the purchase of houses, state and local governments borrow and businesses get enough short-term cash to make payroll.
"The pressure is now on Congress to get its act together and provide the support that the Fed cannot do — helping the vulnerable people who face the biggest health and economic consequences," said James Knightley of ING in a report.
"The risk is that this wall of support from the Fed and the positive reaction in markets may give Congress a sense that it has more time and the pressure to deliver a package is reduced," Knightley said.
As Congress was locked in stalemate, the number of known infections worldwide jumped past 380,000. After just a few weeks, the United States has more than 46,000 cases and more than 600 deaths.
For most people, the coronavirus causes only mild or moderate symptoms, such as fever and cough. Those with mild illness recover in about two weeks. Severe illness including pneumonia can occur, especially in the elderly and people with existing health problems. Recovery could take six weeks in such cases.
Also Monday, trading on the New York Stock Exchange went all-electronic for the first time after the trading floor was temporarily closed as a precaution. The exchange announced the move last week after two employees tested positive for the virus. The number of floor traders had dwindled sharply in recent years as more trading become electronic.
Wall Street and some other stock markets have lost nearly one-third of their value over the past month as business shutdowns spread and airlines, retailers and other industries suffer rising losses.
Economists increasingly say a recession seems inevitable. Analysts are slashing their forecasts for upcoming corporate profits. Forecasters say they cannot project how deep the downturn might be or how long it will last.
Professional traders say investors need to see a decline in numbers of new infections before markets can find a bottom.
Congress debated through the weekend on the rescue plan, but White House officials and congressional leaders are struggling to finalize it. Democrats blocked a vote to advance the package Monday. They want to steer more assistance for public health and workers.
On Wall Street, the S&P 500 index lost 2.9% and the Dow sank 3% in another day of sudden swings.
In energy markets, benchmark U.S. crude rose $1.46 to $24.82 per barrel in electronic trading on the New York Mercantile Exchange. Brent crude, used to price international oils, added $1.47 to $30.76 per barrel in London.
The dollar declined to 110.64 Japanese yen from Monday's 111.23 yen. The euro gained to $1.0850 from $1.0721.