Ukraine’s defence ministry likens the explosion to the sinking of a Russian missile cruiser in April.
A large fire on the only crossing between the occupied Crimean peninsula and Russia was caused by a lorry explosion, Russian officials say.
A blast on the road section led to oil tankers on the rail section catching fire, before the road collapsed.
Crimea was annexed in 2014 by Russia, which now uses the bridge to move military equipment into Ukraine.
The official Twitter account of the Ukraine government responded to the fire by tweeting: “Sick burn.”
An adviser to Ukraine’s President Zelensky, Mykhailo Podolyak, called the damage a “beginning” – but did not directly claim Ukrainian responsibility.
“Everything illegal must be destroyed, everything stolen must be returned to Ukraine, everything occupied by Russia must be expelled,” he tweeted.
Meanwhile, the Ukraine defence ministry compared the bridge explosion to the sinking of Russia’s Moskva missile cruiser in April.
“Two notorious symbols of Russian power in Ukrainian Crimea have gone down,” it tweeted. “What’s next in line?”
It is hard to exaggerate the significance, and symbolism, of seeing the Kerch bridge on fire. Opened by President Putin in 2018, it was meant to symbolise that Crimea was Russian.
Russia has used the bridge to move military equipment, ammunition, and personnel from Russia to battlefields in southern Ukraine.
As such, Ukrainian authorities said it was a legitimate target, as they vow to retake the peninsula.
Any attack on Crimea, where the Russian army has a massive presence, will be seen as another massive humiliation for the Kremlin.
The bridge is particularly hated by Ukrainians. Social media in Ukraine erupted in celebration on seeing the fire – one day after Russian President Vladimir Putin turned 70.
Russia’s National Anti-Terrorism Committee (NAC) said: “At 06:07 Moscow time today [03:07 GMT], an explosion was set off at a cargo vehicle on the motorway part of the Crimean bridge on the side of the Taman peninsula, which set fire to seven fuel tanks of a train that was en route to the Crimean peninsula.
“Two motorway sections of the bridge partially collapsed.”
Crimean parliamentary speaker Vladimir Konstantinov blamed the explosion on “Ukrainian vandals, who have finally managed to reach their bloody hands to the Crimean bridge”.
He added the damage to the bridge would be “promptly restored, since it is not of a serious nature”.
President Putin has been briefed about the “emergency” on the bridge and has ordered a government inquiry, Kremlin spokesman Dmitry Peskov said, in comments quoted by Interfax news agency.
The 19km (12-mile) bridge across the Kerch Strait, which cost £2.7bn to build, was opened by President Putin four years after Moscow illegally annexed Crimea.
The crossing is more than 100 miles from Ukrainian-held territory. One explosives expert told the BBC the fire was probably not caused by a missile.
“The lack of obvious blast / fragmentation damage on the road surface suggests that an air-delivered weapon was not used,” he said.
He said it was possible that “a well-planned attack from below may have been the cause”.
“I suspect explosives on the road bridge and train deck were initiated near simultaneously using coded radio command,” he added.
Kyiv has the momentum in this conflict. The army has reclaimed large swathes of territory, forcing Russian troops to abandon long-held positions.
Amid the losses, Moscow has begun a chaotic military mobilisation – which led to rare anti-war protests in Russia, and a huge exodus of military-age men.