"The PMI is indicating euro area GDP growth close to 0.4 percent in the third quarter, a solid albeit unspectacular rate of expansion," said Chris Williamson, chief economist at survey compiler Markit.
"Policymakers have little scope for complacency, however, as slower growth in the emerging markets and recent financial market volatility, as well as a stronger euro, have the potential to hit the economy's performance in coming months."
Markit's final August Composite Purchasing Managers' Index (PMI) beat an earlier estimate of 54.1, settling at 54.3 -- its highest level since May 2011. In July it registered 53.9 and has now been above 50, which denotes expansion, since July 2013.
The PMI for the bloc's dominant service industry rose to 54.4 from July's 54.0. The flash estimate was 54.3. The manufacturing PMI, released on Tuesday, dipped to 52.3 from 52.4.
An earlier composite PMI from Germany, Europe's largest economy, leapt to 55.0 from 53.7 in July, smashing a flash estimate for a more modest rise to 54.0. Italy's composite reading rose to 55.0, its highest since March 2011.
Spain's PMI also soared but it was a different story in France where the composite PMI slumped to 50.2, its lowest since the start of the year.
In another positive sign for the ECB, firms increased prices last month -- although only barely -- for the first time since early 2012. The composite output price sub index climbed to 50.1 from July's 49.8.
The ECB began pumping 60 billion euros a month of fresh cash into the economy about half a year ago through bond purchases with the aim to boost inflation. But official data showed prices rose just 0.2 percent in the 19-country bloc last month.
To meet the upturn in demand, service firms took on staff at the second fastest rate since May 2011. The index rose to 52.3 from 51.7 in July. Unemployment surprisingly fell to a more than three-year low of 10.9 percent in July.