The rate dropped to 9.1% last month, from a downwardly revised 9.2% in May.
Separately, inflation remained unchanged in July at 1.3%, according to a preliminary estimate from Eurostat, the European Union's statistics office.
However, core inflation, which excludes food, alcohol and energy prices, rose to 1.2% from 1.1% in June.
Inflation in the 19-nation bloc still remains well below the European Central Bank's target of close to, but below, 2%.
Unemployment fell in all 19 eurozone member countries except Estonia.
The lowest unemployment rates last month were in Germany at 3.8% and Malta at 4.1%.
Greece has the highest rate of unemployment in the eurozone at 21.7%, and also the highest rate of youth unemployment at 45.5%, although those figures refer to April - the latest data available for the country.
The second highest unemployment rate was 17.1% in Spain, which was down from 19.9% a year earlier.
Spain also had the second highest level of youth unemployment among 15-24 year olds at 39.2%.
The increase in the core rate of inflation had not been expected by analysts.
Signs of recovery in the eurozone have led to much speculation that the European Central Bank (ECB) will soon start to cut back its 2.3 trillion euro (Â£2 trillion) quantitative easing (QE) programme, which has involved bond buying to try to boost the bloc's economy and avoid deflation.
ECB president Mario Draghi has previously said that sluggish core inflation and wage growth are reasons to be cautious about reining in the policy.
"Today's upside surprise in core inflation is likely to give the ECB some comfort, even though its level remains low," Morgan Stanley economist Daniele Antonucci said. "We expect a QE tapering announcement this autumn."
However, Connor Campbell, financial analyst at Spreadex, said the markets felt the latest data was not  strong enough "to push the ECB in a hawkish direction", and tighten its policy.
The unemployment rate in the eurozone has fallen to its lowest since February 2009, according to the latest official figures.