Japan’s Nikkei 225 index rose 1.3%, Hong Kong’s Hang Seng jumped 2% and Australia’s ASX 200 rose to a five-year high after the announcement.
Stock exchanges in mainland China, Taiwan and South Korea were closed for public holidays.
In the US, the three main stock indexes closed at record highs on Wednesday.
Many investors had speculated that the Federal Reserve would begin reducing its $85bn (£54bn) bond-buying plan this month.
But in a statement released after its two-day policy meeting, the Fed said there was no fixed timetable for it to begin scaling back – or “tapering” – its stimulus.
The central bank said it was taking a more cautious stance because of an “elevated” unemployment rate and concerns about the US economic recovery.
“The committee decided to await more evidence that progress will be sustained before adjusting the pace of its purchases,” it said.
The Fed also cut its forecast for growth this year to between 2.0% and 2.3%. That compares to a June estimate of between 2.3% and 2.6%.
Gold had its biggest one-day jump in four years, rising by more than 4% to $1,364 per ounce following the Fed decision.
But it is emerging Asian stocks and currencies that have posted the largest gains.
Indonesian stocks have surged as much as 7%, Philippines shares rose 3.6%, and Thailand’s benchmark index gained nearly 2%.
The Malaysian ringgit and Thai baht have both strengthened by nearly 2%.
Analysts said Wednesday’s rally was expected, but added that questions remained over whether the gains were sustainable.
“Emerging markets will be the main beneficiaries, given their sensitivity to the Fed policy, which has been fairly evident,” Kelly Teoh from IG Markets said.
“I don’t think it’s good to continuously have stimulus in the markets. We saw in May – when the Fed first talked about the taper – how violent the sell-off can become.”
“The stimulus masked the underlying growth and structural issues in emerging Asian markets. If you look at the current account deficits you can see that growth has been stalling,” Ms Teoh said.
“But this all just being swept under the carpet now.”
Earlier this year, growing expectations of a Fed tapering sparked a sharp sell-off in emerging markets as investors moved their money out of the country.
The Fed’s programme, which is also known as quantitative easing, meant that large sums of extra money was invested in Asian financial markets because many offered high returns.
Concerns that this would be reduced have caused investors to move their money out of these countries and into other assets.
As a result, emerging market currencies such as the Indian rupee and the Indonesian rupiah saw their values depreciate significantly.
Market participants are now turning their eye towards the Fed’s next policy meeting in October.
Ms Teoh said it unlikely that policymakers will pare the stimulus then either.
But she said it was important the Fed ultimately end its asset programme soon because markets are operating on a “borrowed tab”.
“It is quite evident that QE does not help the US labour market. It’s been five years and people are still giving up hope about finding a job. It helps the stock market but it doesn’t trickle down or its very slow.”
David Carbon, chief economist at Singapore bank DBS also believes it is unlikely the Fed will announce a taper at their next meeting in October either.
He said weak employment growth, low inflation, slowing private sector growth and higher long-term interest rates in the US would make the Fed “sit tight”.
“The data are key. And they’re not looking like October at the moment,” he said.