Police in the principality had opened an attempted murder investigation into Monday's incident but were not qualifying it as a terrorism investigation, chief prosecutor Stephane Thibault told reporters.
The motive remains unclear.
One of the three injured was a woman in life-threatening condition, he said.
The other was a man who was no longer in life-threatening condition and a child whose life was not in danger, he said.
The suspected attacker fled into neighbouring France, authorities have said.
Media reports identified Ukrainian construction tycoon Vadym Yermolaiev as being among the injured.
Ukrainian news site Ukrainska Pravda said he was targeted by Ukrainian sanctions in 2023 for ties to Russia.
The injured woman was being treated at a hospital in Nice, French news broadcaster LCI reported, and her partner and a 13-year-old child suffered less severe injuries but remained at hospital, he said.
The explosion occurred on Monday night at the entrance of a residence near the French border.
French and Monaco authorities were searching for an unidentified suspect, whose motive was under investigation, authorities said.
The three victims were "apparently returning home peacefully" in the early evening, according to surveillance footage, said Christophe Mirmand, the minister of state for Monaco.
"They were caught in the explosion as they crossed the threshold of their apartment building," he said.
The victims were "regular" residents of Monaco, but authorities did not yet know whether the family had been threatened in the past, Mirmand said.
"It appears that the family was specifically targeted," he said, noting the alleged perpetrator "had walked around the area several times while waiting for the victims", according to surveillance footage.
The attack has shocked the elite principality on the Mediterranean Coast.
Monaco's Prince Albert II described it as "an odious act" and said all the country's services were mobilised to ensure security.
A French national police official said a search was under way for the suspect.
Yermolaiev, a Ukrainian-born businessman from the city of Dnipro, became one of the country's best-known property developers and has regularly appeared in rankings of Ukraine's wealthiest businesspeople.
In an interview with Forbes Ukraine, Yermolaiev said he renounced his Ukrainian citizenship and became a Cypriot citizen in 2017.
In December 2023, Ukrainian President Volodymyr Zelenskiy imposed sanctions on Yermolaiev as part of a broader package targeting individuals and companies Kyiv said had business links to Russia or Russian-occupied territories.
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