The eurozone's economy grew by 0.3% in the final three months of 2013, up from 0.1% growth in the previous quarter.
It was the third quarter of growth since the end of an 18-month recession, the longest period of contraction to affect the single currency area.
The eurozone figures include 17 of the EU's economies. Latvia became the currency zone's 18th member in January.
Across the whole 28-nation EU, including the UK, growth for the October-to-December period was 0.4%.
The figures from Eurostat, the EU's statistics office, also showed that during 2013, GDP contracted by 0.4% in the eurozone, but increased by 0.1% in the EU as a whole.
"The eurozone's recovery has moved up a gear," said Chris Williamson, chief economist of Markit.
"Not only has the pace of growth picked up to the fastest since the second quarter of 2011, but the recovery is also becoming more broad-based, encompassing core and so-called 'periphery' countries alike."
Earlier, French government figures indicated the country's economy grew by 0.3% in the last three months of 2013.
The INSEE statistical office also reported that growth was zero in the third quarter of 2013, revised up from an initial estimate of a 0.1% contraction.
The figures mean that the world's fifth-largest economy escaped falling back into recession.
Over the whole of 2013, the French economy grew by 0.3%.
The German economy also notched up higher growth in the October-to-December period.
The country's GDP expanded by 0.4% in the final quarter of 2013, after seeing growth of 0.3% in the previous three months, according to the federal statistics office, Destatis.
Destatis said the figures were boosted by exports and capital investment, but there were "mixed signals" from domestic demand, with a drop in household spending.
According to preliminary figures, Germany's economy grew by 1.3% in 2013, the statistics office said.
In general, the German figures were better than analysts had been expecting.
"Germany remains the economic stronghold of the eurozone," said economist Carsten Brzeski of ING.
"In fact, this morning's data was one of those positive surprises the eurozone has seen too seldom over the last few quarters. Let's hope it won't be the last one."
Italy's official statistics office also issued figures showing that its economy returned to growth after a two-year recession.
Istat said GDP grew by 0.1% in the final quarter of 2013, after showing zero growth in the previous three months.
However, during 2013 as a whole, the economy shrank by 1.9%.