"They tried to ruin the fair, but four spoilt brats won't ruin the Expo," Renzi told Italian public television, while saluting the security forces.
On Friday afternoon, hours after the Expo 2015 opened, dozens of protesters opposed to Milan's hosting of the fair took to the streets wearing balaclavas or gas masks.
They smashed windows and torched several cars and bins during a "No Expo" march which organisers said had brought 30,000 people onto the streets.
Police responded to the violence firing tear gas. Water cannons were used to put out the blazes, including one at a bank.
"Those who seek a negative climate of uncertainty and violence at an event that inspires peace, dialogue... and citizenship have lost," Agriculture Minister Maurizio Martina told reporters.
Organisers said "a great many" people visited the fair on Friday despite the clashes, but did not confirm the Italian media's estimate of 200,000.
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Eleven million entry tickets have already been sold for the agriculture and gastronomy-themed event that will run until October 31, according to Expo 2015 commissioner Giuseppe Sala.
Milan prosecutors have opened an investigation into Friday's violence, and the authorities have pledged 1.5 million euros ($1.7 million) to help those whose cars or property was damaged in the clashes.