Regional governor Oleh Kiper, writing on the Telegram messaging app, said four of the injured were in serious condition and receiving urgent treatment.
He said the injured included two children and a pregnant woman.
Pictures posted online showed an ornate building close to the seafront ablaze and smoke billowing skyward.
Video footage, which could not be immediately verified, showed people receiving treatment on the street alongside pools of blood.
Ukrainian navy spokesman Dmytro Pletenchuk, in a posting on a military Telegram channel, said the strike was conducted by an Iskander-M ballistic missile with a cluster warhead.
These missiles are harder to intercept.
Public broadcaster Suspilne said the roof of the building, described as a law academy, had been nearly destroyed.
I have just received reports on the situation in Odesa following a Russian missile strike and in Kharkiv following a Russian guided aerial bomb strike.— Volodymyr Zelenskyy / Володимир Зеленський (@ZelenskyyUa) Unfortunately, a number of people have been killed in Odesa. My condolences to their close ones. Many people have been injured,… pic.twitter.com/Dg7eAEOqdtApril 29, 2024
It said the academy's president, a prominent former member of parliament, Serhiy Kivalov, was among the injured.
Odesa has been a frequent target of Russian missile and drone attacks, particular port infrastructure.
Ukrainian President Volodymyr Zelenskiy said on Monday that US weapons were starting to arrive in Ukraine in small amounts but deliveries needed to be faster as advancing Russian invasion forces were trying to take advantage.
Zelenskiy told a joint press conference in Kyiv alongside NATO Secretary-General Jens Stoltenberg that the situation on the battlefield directly depended on the speed of ammunition supplies to Ukraine.
"Timely support for our army. Today I don't see anything positive on this point yet. There are supplies, they have slightly begun, this process needs to be sped up," he said.
The United States approved a $US61 billion ($A93 billion) aid package last week, ending six months of congressional deadlock and raising Ukrainian hopes that its critically low stocks of artillery shells will soon be replenished.
Ukrainian President Volodymyr Zelenskiy has held talks with NATO Secretary-General Jens Stoltenberg. (AP PHOTO)
Stoltenberg, who held talks with Zelenskiy during his third wartime visit to Kyiv, told Ukrainians that NATO members had failed to live up to their promises of military aid in recent months but that the flow of arms and ammunition would increase.
"I will ... be very honest with President Zelenskiy and also with the Rada (Ukrainian parliament) that NATO allies have not delivered what we have promised over the last months," Stoltenberg said on the train taking him into Kyiv on Monday.
The NATO chief pointed to the United States aid bill and an announcement last week by United Kingdom Prime Minister Rishi Sunak of a "record high" commitment to Ukraine.
He also noted Germany had agreed to provide another Patriot air defence system to Ukraine while the Netherlands had boosted its military aid to Ukraine.
He said he expected other "new commitments to come".
"This will make a difference - as the lack of support made a difference," he said, alluding to Ukrainian setbacks on the battlefield over the past few months.
Russia has said a renewed influx of US weapons will not change the situation on front lines and that it will target storage sites holding foreign weaponry in Ukraine.
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