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.
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