The dollar was a major beneficiary, surging to 104.15 yen while the euro toppled back to $1.3685.
The broad-based slide in the yen was viewed as positive for Japanese exports and profits, and thus for the Nikkei which climbed 1.7% to threaten its peak for this year.
Stocks in Sydney and Seoul both gained 0.8% while MSCI's broadest index of Asia-Pacific shares outside Japan edged up 0.2%.
After months of agonising, investors took the Fed's decision to trim its bond buying by $10 billion to $75 billion a month as a modest step and one the US economy could well withstand.
Crucially, the Fed softened the blow by making its forward guidance even more dovish.
"It likely will be appropriate to maintain the current target range for the federal funds rate well past the time that the unemployment rate declines below 6-1/2%, especially if projected inflation continues to run below the committee's 2% longer-run goal," the Fed statement said.
Alan Ruskin, global head of G10 currency strategy at Deutsche Bank in New York, noted the Fed's forecasts for the funds rate had also been trimmed out to the end of 2016.
"This is a very dovish taper-lite where the Fed has done its utmost to provide an offset with forward guidance," said Ruskin.
"It tends to elevate the importance of the inflation rate in decision making should it be meaningfully undershooting target, which is very constructive for risky assets."
The market seemed to agree, with the Dow ending Wednesday up 1.84%, while the S&P 500 gained 1.66% and the Nasdaq 1.15%.
European markets had rallied earlier after think-tank Ifo reported German business morale in December was at its highest since April 2012.
LOW FOR LONGER
The Fed's message that tapering was not tightening looked to have resonated in debt markets as Fed fund futures firmed all the way out to the early 2016 contracts. A first hike in the funds rate is not fully priced in until November 2015.
Treasury yields were stable for three years ahead, while rising at the longer end as the yield curve steepened. Yields on 10-year notes increased 5 basis points to 2.89%, but remain below their 2013 peak of 3%.
Still, tapering could be a double-edged sword for some Asian countries since it could accelerate the "great rotation" of funds out of emerging markets and into developed world assets.
Indonesia, the Philippines, Thailand and India have all been hit to a varying extent in recent months, though some are now better prepared for the change.
Notably the mood around India has improved enough that the country's central bank could hold off on hiking interest rates on Wednesday, surprising many.
In commodity markets, gold was the main casualty of the Fed move, and the resulting jump in the dollar. After gyrating wildly, spot prices for the metal were off at $1,220.91 an ounce and uncomfortably close to the year low at $1,211.80.
Oil markets shrugged off the Fed's decision, perhaps encouraged by signs of a pick up in global growth.
Brent crude ended Wednesday up 84 cents at $109.24 a barrel. US oil futures edged back 19 cents to $97.87 a barrel but are still up over a dollar on the week so far.