(Read more: Euro zone optimism wanes)
It comes after the region posted growth of just 0.1 percent in the third quarter - a significant slowdown from an expansion of 0.3 percent in the second three months of the year.
Over 2013 as a whole, however, the euro zone's economy still contracted by 0.4 percent.
Howard Archer, chief U.K. and European economist, said that while this level of growth was "far from dynamic," it was a step in the right direction.
Daiwa Capital Markets' Head of Economic Research Chris Scicluna described 0.3 percent growth as "strong as might reasonably have been hoped."
Bridging the gap?
On a country level, the German and French economies grew at a similar pace in the final quarter of 2013, bridging the gap seen in recent quarters between the euro zone's two largest economies, as investment picked up in both regions.
France's economy grew 0.3 percent in the final quarter of 2013, slightly more than the 0.2 percent analysts had expected, after seeing no growth in the third quarter. Germany grew 0.4 percent in the quarter, a faster growth rate than the 0.3 percent growth consensus.
Meanwhile Italy's fourth-quarter gross domestic product (GDP) came in at 0.1 percent on the quarter, meeting expectations, and Spain's growth picked up to 0.3 percent.
One of the themes of the euro zone's recovery - since it exited its longest recession in over 40 years in the second quarter of 2013 - has been a divergence between the performance of the region's economies.
These latest GDP figures, however, add to signs that the gap could be narrowing.
(Read more: Euro soars as Draghi plays down deflation talk)
The French statistics agency INSEE also revised its figures for the third quarter from a contraction of 0.1 percent to gross domestic product of 0 percent.
The return to growth for the French economy means the nation has managed to escape falling back into recession. Finance minister Pierre Moscovici said "growth of 0.3 percent for the whole of 2013: an encouraging number" on social media network Twitter.
While better than expected, the weakness of France's recovery is adding to pressure on President Francois Hollande to deliver faster growth. The unpopular socialist premier has embarked on a shift to more business-friendly policies to bring down near-record unemployment, which is close to 11 percent. Hollande is counting on a planned 30 billion euro corporate tax break to ensure France does not fall too far behind other euro zone economies that are recovering faster.
The German Statistics Office said growth had picked up unexpectedly thanks to a rise in exports in 2013, as exports rose a lot more than imports.
Analysts were surprised by the acceleration in German economic growth, adding the positive data was something the euro zone had not seen very much of over the last few quarters.
(Read more: Euro zone GDP seen weak as optimism wanes)
"Is the German economy finally picking up? Recent monthly data had painted a rather confusing picture with strong soft data but disappointing hard data. Today's growth outcome is actually better than monthly hard data had suggested," said ING's Carsten Brzeski.