The MSCI world equity index, which tracks shares in 49 countries, was up 0.1% on the day at 1101 GMT, having recovered from the one-month low it hit in the aftermath of the Fed's meeting.
The STOXX 600 was 0.3% lower on the day but was up around 1.5% from the lows it hit on Monday.
But Wall Street futures pointed to a slightly higher open in the U.S.
"The market’s still digesting the Fed news," said Mo Kazmi, portfolio manager and macro strategist at UBP.
"I think a lot of that move was exacerbated by stretched positioning and now what we're seeing is perhaps reflation trades being put back on and the market normalising to some extent, realising that for now it’s just a subtle shift from the Fed."
Powell's comments helped the yield on benchmark 10-year U.S. Treasuries lower and put the brakes on a rising U.S. dollar.
The 10-year U.S. Treasury yield was at 1.4767% at 0801 GMT.
The U.S. dollar slipped as European markets opened. The euro was steady against the greenback at $1.1943.
Early PMI data showed that euro zone business growth accelerated at its fastest pace in 15 years in June as the easing of more lockdown measures and the unleashing of pent-up demand drove a boom in the bloc's dominant services industry.
Germany's private sector growth was also lifted to its highest level in more than a decade in June, the PMI survey showed. In France, business activity edged higher, but not as much as expected.
In Britain, growth in the private sector cooled slightly from the all-time high hit in May, but inflation pressures faced by firms hit record levels. The Bank of England meets on Thursday.
Berenberg economists Holger Schmieding and Kallum Pickering wrote in a note to clients that the euro zone economy is likely to recover to its pre-pandemic level of GDP in Q4 2021, while for Britain it will be Q1 2022.
UBP's Kazmi said that he is positioned for higher yields in Europe, as it overtakes the United States in terms of vaccinations, lockdown easing and economic recovery from COVID-19.
"It will be interesting to see if the German Bund can follow the U.S. rate move with yields moving higher in Europe – it is something that we think could happen," he said.
"The fact that the Fed has moved more hawkishly will allow the ECB to be more comfortable perhaps in moving more hawkish, or less dovish, over time."
Germany's benchmark Bund yield was steady at -0.176% at 1201 GMT.
Oil prices jumped to their highest in more than two years.
Elsewhere, bitcoin was up around 5% on the day, above the $34,000 mark. The cryptocurrency dropped to as low as $28,600 on Tuesday - its lowest since January. Ether was trading around $2,000.
(Reporting by Elizabeth Howcroft; Editing by Emelia Sithole-Matarise and Angus MacSwan) ((Elizabeth.Howcroft@thomsonreuters.com; +44 02075427104;))