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Financial spreadbetters saw the momentum carrying on into Europe, with Britain's FTSE seen rising as much as 0.5 percent at the open, Germany's DAX up 0.6 percent and France's CAC 0.7 percent higher.
MSCI's broadest index of Asia-Pacific shares outside Japan gained 0.8 percent.
Tokyo's Nikkei brushed aside a stronger yen and outperformed the rest of the region, advancing as much as 1.5 percent at one point to a 4-1/2-month high.
"Yesterday's Fed announcements include more than one piece of good news, so that's positive for Japanese market sentiment," said Nobuhiko Kuramochi, a strategist at Mizuho Securities in Tokyo.
"The Fed sees the U.S. economy as on track, while it hinted of low interest rates in the long term."
On Wall Street, the S&P 500 ended at a record high after the Fed hinted at a slightly faster pace of interest-rate increases starting next year but suggested rates in the long run would be lower than it had indicated previously.
World markets have been buffeted in recent weeks by concerns over China's slowdown and an uneven global recovery. Turmoil in Ukraine and Iraq have further undermined sentiment.
The Fed's accommodative policy stance is seen as one of the positives, as rising consumption in the U.S. is expected to help underpin some of Asia's big export-driven economies.
The dollar struggled against its peers, hurt as U.S. Treasury yields fell sharply on the Fed's long-term rates projection and as policymakers showed little discomfort over recent signs of a pick up in consumer prices.
The benchmark 10-year Treasury note yield fell to as low as 2.575 percent, its lowest in a week.
Elsewhere in sovereign debt, the usually placid Japanese government bond market felt a knock after the Bank of Japan on Wednesday said it may trim the amount of longer-term maturity bonds it purchases regularly from the market.
The small fine-tuning by the BOJ was enough to steepen the yield curve significantly, unsettling a market that has become so dependent on the BOJ's bond purchases designed as a part of its extensive monetary easing programme. [JP/]
The dollar index, a gauge of the greenback's strength against a basket of key currencies, was down 0.3 percent.
The dollar traded little changed at 101.92 yen after shedding 0.2 percent on Wednesday.
The euro was flat at $1.3592, taking a breather following the previous session's 0.35 percent rise.
In commodities, Brent crude hovered near Wednesday's nine-month closing high of $114.26 a barrel hit on persistent worries over oil exports from war-torn Iraq, where Islamic militants seized much of its northern region as Baghdad's forces crumbled. [O/R]
"The oil market remains in high alert, but is in a holding pattern at this stage awaiting further developments in Iraq," said Michael McCarthy, chief strategist at CMC Markets in Sydney.
Platinum and palladium extended their winning streak, with both trading near one-week highs as a miners' strike in major producer South Africa looked set to continue.
Analysts saw the miners' strike curtailing the South African rand's strength, which surged 1 percent against the dollar on Wednesday on news the country's current account deficit narrowed to levels last seen in 2012.
