Moscow said the "mass retaliatory strike" was aimed at a factory producing drones of the type used to attack the bridge between Crimea and the Russian mainland.
The Ukrainian air force said six Kalibr cruise missiles and 36 Shahed attack drones were launched at the country's south and east.
Damage was reported to Odesa's port infrastructure and to private homes after rocket fragments hit the city.
An elderly man was reported to be wounded.
An industrial site was hit in the city of Mykolaiv, the military said, causing a fire that its mayor called "quite serious".
Ukraine suspects the missiles were fired from the Black Sea frigate Admiral Essen and the drones from occupied Crimea. The air force said it intercepted most of the incoming fire.
The barrage comes a day after Russia vowed to retaliate after the explosion on the bridge linking Crimea to the Russian mainland.
Russia said road traffic was partly restored on Tuesday after an attack it blamed on Ukrainian drones.
Moscow's forces "carried out a mass retaliatory strike overnight using precision sea-based weapons against facilities where terrorist acts against the Russian Federation were being prepared using uncrewed boats", the Russian Defence Ministry said.
"In addition, storage facilities holding around 70,000 tons of fuel used to supply the Ukrainian military's equipment were destroyed" near the cities of Mykolaiv and Odesa, it said.
The Kremlin announced on Monday that it would not extend the grain deal that provided for safe shipping corridors in the Black Sea.
Ukraine is one of the world's top agricultural exporters and Russia's decision brought fears for global food security.
Tuesday's attack on Odesa shows that Russia "wants to endanger the lives of 400 million people in various countries that depend on Ukrainian food exports", said Ukrainian presidential adviser Andriy Yermak.
Russia said safety guarantees for shipping were no longer in place, with part of the Black Sea now designated as dangerous for mariners.
Russia on Tuesday issued a veiled warning over the future of grain exports through the Black Sea.
"Without appropriate security guarantees, certain risks arise here [in the Black Sea]," Kremlin spokesman Dmitry Peskov said.
If a new arrangement to allow for exports were "formalised without Russia, then these risks should be taken into account", he said.
Mr Peskov said Ukraine was using the Black Sea export corridor "for combat purposes".
Ukraine said it hoped to keep the sea corridor open with help from Turkey and the UN, who brokered the original deal.
"Ukraine, the UN and Turkey can jointly ensure the operation of the food corridor and the inspection of vessels," President Volodymyr Zelenskyy said overnight.
"It is necessary for everyone in the world, and everyone who supports it will become a saviour of life in a huge territory from Morocco to China, from Indonesia to Lebanon."
Kyiv is also looking at expanding the volume of grain that can be carried through other routes, said Dmytro Senik, Ukraine's ambassador to the UAE.