Russian President Vladimir Putin has accused Ukraine of attacking the bridge to Russian-annexed Crimea, saying that it was an "act of terrorism".
President Putin said Ukraine's intelligence forces had aimed to destroy a critically important piece of Russia's civil infrastructure.
He was speaking at a meeting with the head of the Investigative Committee of Russia, Alexander Bastrykin.
Officials say three people were killed in the blast on the bridge.
The victims were in a nearby car when a lorry blew up, Russian officials say.
"There is no doubt, this is an act of terrorism aimed at destroying Russia's critical civilian infrastructure," Mr Putin said.
"Its authors, perpetrators and beneficiaries are the security services of Ukraine."
Mr Bastrykin said that citizens of Russia and some foreign states had aided preparations for the attack.
According to Mr Bastrykin, investigators have established that the truck which they say blew up travelled through Bulgaria, Georgia, Armenia, North Ossetia and Krasnodar Territory.
He has ordered an investigation into the incident which brought down sections of the roadway.
Ukrainian officials have not indicated that their forces were behind the attack.
But an adviser to Ukraine's President Volodymyr Zelensky, Mykhailo Podolyak, denied Mr Putin's accusation.
He wrote that there is "only one terrorist state here" and that the "whole world knows who it is".
"Does Putin accuse Ukraine of terrorism? It looks too cynical even for Russia," he said.
On Saturday, President Volodymyr Zelensky acknowledged the incident in his nightly address, saying: "Today was not a bad day and mostly sunny on our state's territory."
"Unfortunately, it was cloudy in Crimea. Although it was also warm," he added.
Russian authorities partially re-opened the roadway part of the bridge hours after the attack but for light traffic only.
The railway part of the bridge - where oil tankers caught fire - has also reopened.
The 19km (12-mile) bridge, the longest in Europe, is an important supply route for Russian forces fighting in Ukraine.
Russia has used the bridge to move military equipment, ammunition, and personnel from Russia to battlefields in southern Ukraine.
It was opened by Mr Putin in 2018, four years after Russia's annexation of Crimea.
Security camera footage released on social media showed a truck - allegedly from the Russian city of Krasnodar, an hour's drive from the crossing - moving west across the bridge at the time of the explosion.
The footage shows a huge fireball erupting just behind - and to one side - of the truck as it begins to climb an elevated section of the bridge.
The speed with which the truck bomb theory started to spread in Russian circles was suspicious. It suggested the Kremlin preferred an act of terrorism to a more alarming possibility: that this was an audacious act of sabotage carried out by Ukraine.
"I've seen plenty of large vehicle-borne IEDs [improvised explosive devices] in my time," a former British army explosives expert told me. "This does not look like one."
A more plausible explanation, he said, is a massive explosion below the bridge - probably delivered using some kind of clandestine maritime drone.
"Bridges are generally designed to resist downwards loads on the deck and a certain amount of side loading from the wind," he said. "They are not generally engineered to resist upward loads. I think this fact was exploited in the Ukrainian attack."
Some observers have noted that in one of the other security camera videos, something that looks like the bow wave of a small boat appears next to one of the bridge supports, a split second before the explosion.