The landmark work, tucked away in a little square near the Pantheon, features an elephant carrying the obelisk on its back and was first placed in the Piazza della Minerva in the 17th Century.
Gian Lorenzo Bernini oversaw the sculpture of the elephant, which had the tip of its left tusk broken off in the overnight incident. The fragment was found on the ground beside the statue. Virginia Raggi, the mayor of Rome, said the breakage was "painful for all Romans".
"The breakage occurred in a place where a restoration had already taken place," Rome's councillor for culture, Luca Bergamo said, explaining that it was not yet clear if the damage had been deliberate.
Italy's Culture Minister Dario Franceschini said the incident showed a need for more video surveillance of historic landmarks, and harsher punishments for vandals. He added: "It's right that these masterpieces should be in public squares."
The elephant was commissioned by the then pope, Alexander VII, to support an obelisk from ancient Egypt that had only recently been excavated.
The damage to the Bernini elephant comes after fans of Dutch football club Feyenoord caused outrage in February 2015 by damaging a fountain created by the sculptor that stands at the bottom of Rome's fabled Spanish Steps.