The UK's FTSE 100 index rose 1.5%, while the main share indexes in France and Germany saw even bigger increases.
The dollar also rallied against other currencies after the US Federal Reserve increased the range for its benchmark rate to between 0.25% and 0.5%.
The Fed said the rise was part of a "gradual" process to get rates back to normal after years of being near zero.
London's FTSE 100 rose 1.5% to 6,152.86, while Frankfurt's Dax jumped more than 3% and the Cac 40 in Paris was 2.4% higher.
'Christmas has come early'
The European stock markets were following the lead given by markets in the US and Asia.
On Wall Street, the Dow Jones closed up 224.18 points, or 1.3%, at 17,749.09, while in Japan, the benchmark Nikkei 225 closed up 1.6% at 19,353.56.
"With the Dow rising steadily from the moment [Fed chairwoman Janet Yellen] first opened her mouth, the rosy picture she painted of the US economy and the absence of major overseas threats has sent markets surging with relief," said Robert Craig, private client investment manager at MB Capital.
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