The CIPS/Markit composite purchasing managers' index (PMI) rose to 54.1, compared with 53.6 a month earlier - its highest reading in 49 months.
A reading above 50 indicates growth, while a reading below 50 suggests a fall in activity.
Markit said the services sector had seen its best quarter for four years.
In addition, factories enjoyed their best quarter of production growth for a year, "highlighting the broad-based nature of the upturn".
Employment and new orders also rose at their strongest rates for four years in the second quarter.
"Despite the cloud of the Greek debt crisis hanging over the region, the eurozone saw economic growth accelerate to a four-year high in June," said Chris Williamson, chief economist at Markit.
"The PMI is signalling GDP growth of 0.4% for the region as a whole in the second quarter."
The eurozone's gross domestic product (GDP) rose by 0.4% in the first quarter, according to official figures.
Mr Williamson added that the eurozone's economy was on course to grow by 2% this year, "though much of course depends on the outcome of the Greek debt negotiations and any resulting impact on growth in the second half of the year".
Business activity picked up in both Germany and France in June, but Germany saw a weaker growth rate in the second quarter compared with the first.
Excluding France and Germany, the rest of the eurozone recorded its best performance for eight years, Markit said.
The eurozone's economic recovery is continuing, a survey has suggested, with business output growing at its fastest rate in four years in June.