In a brief TV address, Renzi did not say how many Italians were among the casualties, but local media said 11 had been taken hostage, only one of whom had escaped.
"We are like a family which has suffered a painful loss," Renzi said.
The attackers who believed they were "destroying our values" would not get a drop of encouragement from Italy, he said.
"We are stronger," Renzi said.
"The terrorists want to rip away the daily fabric of our lives. Our duty is to reply with even greater force, by affirming our values, the values of freedom of which we are proud, and which are stronger than hatred or terror."
Many of the victims were hacked to death.
The Bangladesh military said most of the fatalities were either Italian or Japanese. Thirteen hostages were rescued.