However the report also highlighted continuing problems for France, lagging further behind powerhouse Germany which goes from strength to strength, analysts said.
“Relatively reassuring for the Eurozone overall but certainly not for France,” said Howard Archer at IHS Global Insight.
Markit Economics said its Eurozone Composite Purchasing Managers Index (PMI) for May slipped to 53.9 points from 54 in April.
The composite indicator was at its second-highest level of the past three years, and the average for the second quarter so far was running at its best since the three months to June 2011 period, Markit said.
Germany was higher at 56.1 points while France fell to 49.3 points, back under again the boom-bust line of 50.
The May services sector PMI hit a 35-month high of 53.5 points, up from 53.1 in April, while the manufacturing PMI fell to 52.5 points, a six-month low, from 53.4 in April.
The report was positive after overall economic growth in the 18-nation eurozone slowed unexpectedly to 0.2 percent in the first quarter, well short of forecasts for 0.4 percent.
Markit chief economist Chris Williamson said the slight PMI easing in May “doesn’t change the picture of a region that’s enjoying its best spell of growth for three years.”
The figures suggest the eurozone economy could grow “0.5 percent in the second quarter after the lacklustre 0.2 percent rise in the first three months,” Williamson said.
However, there were two concerns, he said, deflation pressures, where falling prices undercut demand, and a struggling France.
“Of greatest concern is France, living up to its moniker of ‘sick man of Europe’ by sliding back into contraction as Germany continues to enjoy robust growth” at its best since mid-2007, he said.
Archer said the report was “relatively reassuring news for hopes that (a) modest eurozone economic recovery remains intact.”
However, “the eurozone is clearly not finding it at all easy to build up growth momentum,” he said, adding that the European Central Bank remains under pressure to take fresh stimulus measures at its next meeting in June.