Worries about Greece's debt crisis also eased after rating agency Standard & Poor's removed the threat of an immediate downgrade.
MSCI's all-country world stock index <.MIWD00000PUS> rose half a percent with Europe's FTSEurofirst 300 <.FTEU3> up 0.4 percent around a two-month high.
Earlier Japan's Nikkei <.N225> closed up 1.17 percent while, overnight, stocks on Wall Street hit 17-month highs. Japanese stocks were boosted by the Bank of Japan's decision to loosen monetary policy.
The Fed essentially presented equity investors with the best of all worlds, saying that it was keeping interest rates near zero and would do so for an extended period, but also pointing to increased momentum in the U.S. economy.
There was no new bad news so this is good. I think the market can continue its upward trend to the end of the quarter, said Giuseppe-Guido Amato, strategist at Lang & Schwarz in Frankfurt.
Investors entered the year fearing the gradual withdrawal of loose monetary policy by the world's central banks, but the expected timing of the shift has been pushed back.
Some 70 percent of respondents in a Bank of America Merrill Lynch fund manager poll this month said the Fed would not tighten policy until the fourth quarter at the earliest, with nearly 40 percent seeing it holding off until next year.
Half the respondents said that the European Central Bank would not tighten until 2011.
Euro zone government bonds were flat despite the gains in stocks, with a degree of relief spreading about Greek debt.
The premium investors demand to buy 10-year Greek government bonds rather than euro zone benchmark German Bunds narrowed.
Standard & Poor's affirmed Greece's triple-B plus credit rating on Tuesday and ended its review for a possible downgrade.
Euro zone countries, meanwhile, have agreed on helping Greece if it needs assistance but have not come up with much in the way of details.
On currency markets, the dollar was under pressure as a result of the Fed's low interest rate pledge.
The euro brought $1.3776.
(Additional reporting by Joanne Frearson; editing by John Stonestreet)