Intense repair works are underway in southern regions and Ukraine has imported electricity from three western neighbours, Kyiv added.
In a similar attack on August 30, Russia said its forces destroyed four Ukrainian military boats carrying up to 50 soldiers in the Black Sea.
The bridge connecting Crimea and Russia carries heavy significance for Moscow, both logistically and psychologically, as a key artery for military and civilian supplies.
Shortly after reporting the downing of the two Ukrainian missiles, Aksyonov said Russian air defense had shot down another missile over the Kerch Strait.
The Black Sea port of Novorossiysk hosts the terminus of a pipeline that carries most Kazakh oil exports through Russia.
Crimea, annexed by Russia in 2014, has been targeted by Kyiv throughout Moscow's Ukraine offensive.
The Ukrainian military appeared to confirm it had launched the drone strike claiming through its press service that it had destroyed an oil depot and Russian arms warehouses.
Seventeen drones sent by Ukraine's forces overnight were 'destroyed' and another 11 were 'suppressed' by electronic means, Russia's defence ministry said in a statement.
The extent of the damage was not immediately clear, but Governor Sergei Aksyonov said he expected rail traffic on the bridge to resume within several hours.
The Russian- installed governor of Crimea, Sergei Aksyonov announced via Telegram that during a strike at night time, the bridge was damaged.