Russia has not yet commented. However, Ukraine's government has come under intense pressure from its neighbour, with a build-up of some 100,000 Russian troops near its borders.
Ukraine's SBU security service says in just nine months last year it "neutralised" 1,200 cyber-attacks or incidents.
At the start of Friday's attack, a message on the hacked websites was posted in three languages, Ukrainian, Russian and Polish.
"Ukrainian! All your personal data has been uploaded onto the public internet," the message read. It continued: "This is for your past, your present and your future."
The SBU in Kyiv later said that to its knowledge no personal data had been leaked and no content had been changed. A number of sites had been suspended to prevent the attack from spreading, the government said, before stating that most affected resources had been restored.