President Volodymyr Zelenskiy said more than 1.5 million people in the southern port city and surrounding region had no electricity, and he described the situation as very difficult.
Since October, Moscow has been targeting Ukraine's energy infrastructure with large waves of missile and drone strikes.
The regional administration said people who relied solely on electricity to power their homes should consider leaving. Officials said Russian strikes hit key transmission lines and equipment in the early hours of Saturday.
"According to preliminary forecasts, it will take much more time to restore energy facilities in the Odesa region than after previous attacks," the administration said.
"We are talking not about days, but even weeks and possibly even two to three months," it said in a Facebook post.
Odesa, Ukraine's largest port city, had a population of more than one million before Russia's February 24 invasion.
Kyiv says Russia has launched hundreds of Iranian-made Shahed-136 drones at targets in Ukraine and describes the attacks as war crimes due to their devastating effect on civilian life. Moscow claims they are militarily legitimate.
In a video address, Zelenskiy said there was a significant shortfall in the amount of power being generated.
Ukraine's prosecutor general's office said two power facilities in Odesa region were hit by Shahed-136 drones.
Ukraine's armed forces said on Facebook that 15 drones had been launched against targets in the southern regions of Odesa and Mykolaiv, and 10 had been shot down.
Tehran denies supplying the drones to Moscow. Kyiv and its Western allies say that is a lie.
Britain's defence ministry said on Saturday that it believed Iran's military support for Russia was likely to increase in the coming months, including possible deliveries of ballistic missiles.
