That confounded the median expectation in a Reuters poll for a dip to 56.6.
"It's a very good result and it's broad based. We've got a good pace of growth here. The fact we have maintained this high level in May is great news for second quarter GDP," said Chris Williamson, chief business economist at IHS Markit.
Williamson said the PMI pointed to second quarter GDP growth of 0.7 percent, much faster than the 0.4 percent predicted in a Reuters poll last week.
Official flash data said the bloc's economy grew 0.5 percent in the first quarter.
Buoyant demand meant firms built up backlogs of work at the second fastest rate in over six years. The sub-index rose to 53.3 from 53.0.
Factories across the currency union had a much better May then predicted.
A Reuters poll said the manufacturing PMI would fall to 56.5 but it instead climbed to 57.0 from 56.7, its highest since April 2011.
An index measuring output, which feeds into the composite PMI, rose to 58.4 from 57.9, also the highest since April 2011.