The US central bank concludes its two-day meeting later and is expected to keep interest rates unchanged, despite mixed economic data.
In Australia, the benchmark S&P/ASX 200 closed 1% higher at 5,595.40 points.
The Kospi index in Seoul rebounded from an 11-week low, adding 0.3% to close at 2,034.86 points.
Hong Kong's Hang Seng finished up 0.7% to 26,753.79, while the Shanghai Composite closed up 1.7% to 4,967.90, ending three days of losses.
Investors seemed to have recovered from the new curbs for margin lending and a flood of new listings this week that had dragged down shares.
On Wall Street, both the Dow and S&P 500 ended Tuesday about 0.6% higher and the technology-focused Nasdaq gained 0.5%.
Many analysts think the Fed might increase interest rates in September and will be watching chair Janet Yellen's statements on Wednesday for further clues.
"Asian markets are seeing a relief rally this morning," said Nicholas Teo from CMC Markets.
"This follow-through, however, may only remain tentative at best, as the focus over the next 24 hours will fix firmly on to the US [Federal Open Markets Committee] decision tonight."
Japan's Nikkei 225 closed 0.2% lower at 20,219.27 points and the broader Topix lost 0.4% to 1,633.70 after the country posted disappointing trade data.
Exports rose by 2.4% from a year earlier in May, which was below economist estimates and indicated weak external demand.
Imports fell by 8.7% on an annualised basis because of a fall in energy costs.
As a result, Japan's trade deficit narrowed slightly to 216bn yen ($1.7bn; Â£1.1bn) last month.
However, analysts said the fall in imports is unlikely to be sustained if the yen weakens further and natural gas costs rise.
"The trade deficit should start to creep higher in the second half of the year," Marcel Thieliant from Capital Economics said.
Asian stock markets mostly rose after US markets rebounded in anticipation of the Federal Reserve decision on interest rates.