The S&P 500 registered its biggest daily percentage gain since July 23.
While trading was choppy following the Fed's latest policy statement and comments by Fed Chair Jerome Powell, stocks finished close to where they were before the central bank news.
In its statement, the central bank also suggested interest rate increases may follow more quickly than expected and said overall indicators in the economy "have continued to strengthen."
Stocks began the day higher as concerns eased over a default by China's Evergrande. Evergrande's main unit said it had negotiated a deal with bondholders to settle interest payments on a domestic bond.
Bank shares rose following the Fed news, with the S&P banks index ending up 2.1% on the day, and S&P 500 financials up 1.6% and among the biggest gainers among sectors.
Some strategists viewed the Fed's comments as mixed.
"So they said we're going to probably start to taper, but they haven't said when and haven't said how much, so we're kind of back where we were a day ago," said Paul Nolte, portfolio manager at Kingsview Investment Management in Chicago.
"Those remain open questions," he said. "Also, financial conditions remain very easy, and that's part of the reason why markets aren't going crazy at this point."
The Dow Jones Industrial Average rose 338.48 points, or 1%, to 34,258.32, the S&P 500 gained 41.45 points, or 0.95%, to 4,395.64 and the Nasdaq Composite added 150.45 points, or 1.02%, to 14,896.85.
Apple and other big technology-related names gave the S&P 500 its biggest boost.
On the downside, FedEx Corp tumbled 9.1% after posting a lower quarterly profit and as the delivery firm cut its full-year earnings forecast.
Advancing issues outnumbered declining ones on the NYSE by a 3.88-to-1 ratio; on Nasdaq, a 2.38-to-1 ratio favored advancers.
The S&P 500 posted nine new 52-week highs and eight new lows; the Nasdaq Composite recorded 52 new highs and 66 new lows.
Volume on U.S. exchanges was 9.91 billion shares, compared with the 9.99 billion average for the full session over the last 20 trading days.
(Reporting by Caroline Valetkevitch; additional reporting by Ambar Warrick and Medha Singh in Bengaluru; Editing by Maju Samuel and Lisa Shumaker)
By Caroline Valetkevitch