Australia`s stock exchange was the first major benchmark index in the region to open to the news - and investors cheered. Sydney`s S&P/ASX 200 was up as much as 1.6% in early trade.
Japan`s Nikkei 225 was up 2.15% at 19,460.20 after marking one of its biggest daily gains in more than two months on Wednesday.
The dollar strengthened against the yen and was buying 122.38 yen in Asian trade. A weaker yen is good for Japan`s big exporters. In South Korea, the Kospi index was up 0.25% at 1,974.1.
The Fed`s decision to raise rates was widely expected and analysts said it indicated a degree of optimism for the world`s biggest economy.
"The move signals confidence in the ongoing recovery in the US economy after the great financial crisis," said Shane Oliver, head of investment strategy and chief economist at AMP Capital in Sydney.
"Given ongoing deflationary risks and slow global growth, future Fed hikes are likely to be cautious and gradual," he added.
"With the Fed decision out of the way, global shares are likely to resume their rising trend but with US shares as a relative underperformer."
The Australian dollar initially fell against the US dollar, but recovered later to buy 72.12 US cents. Australia is hoping to see its currency weaken against the dollar in the coming months.
Mr Oliver said rising US interest rates would help maintain downwards pressure on the value of the Australian dollar through 2016.
Christmas cheer
Analysts said the fact that the Fed was able to increase rates meant the US economy was finally in a position to withstand the move.
"The question for many investors now is whether this momentum in equity markets is likely to develop into a full blown "Santa Rally" through to the end of the year," said IG Markets` Angus Nicholls.
"With little further market-moving releases expected over the coming weeks and a still dovish trajectory for future Fed policy there does seem to be a good chance of this happening," he added.
"It is worth noting that historically 16 December has been the seasonal beginning of a "Santa Rally", so its timing with the Fed meeting does seem particularly auspicious to bring some Christmas cheer to those with equity portfolio holdings."
In China
China`s markets were also in positive territory on Thursday, with Hong Kong`s Hang Seng index up 0.83% at 21,881.97 and the Shanghai Composite up 0.95% at 3,549.66.
Early on Thursday, Hong Kong`s central bank raised the base rate it charges through its overnight discount window by 25 basis points to 0.75 points.
"This is a move that is likely to affect property prices - it will become more expensive for home owners to borrow - and comes after the US fed`s move overnight," the BBC`s Asia business correspondent Karishma Vaswani explained.
Other central bank decisions on Thursday will come from the Philippines and Indonesia - though analysts are not expecting them to move.
"They`re caught in a dilemma of facing slower growth - so they need to cut rates - but can`t do that too aggressively because their currencies will weaken," Ms Vaswani said.
