They displayed a photo of Georgia's exiled former president Mikheil Saakashvili with an inscription "I'll be back!" Georgia's Interpress news agency reported.
Georgian President Salome Zurabishvili's website was "attacked by hackers this afternoon," her spokeswoman told AFP.
"Law enforcement agencies are investigating the incident," Sopho Jajanashvili said.
Interpress said the website for Georgia's general jurisdiction courts as well as websites of a number of government agencies, NGOs and media outlets were also hit by cyber attacks on Monday.
Up to 2,000 sites are believed to have been affected.
The attack also affected servers of Georgia's two major broadcasters, Maestro and Imedi TV, temporarily sending the television stations off the air.
Georgia's interior ministry said it had launched an investigation.
"We still haven't full access to our computer systems," Andro Lashkhi, a lawyer for Imedi TV, told journalists.
In 2008, in the run-up to and during the war between Russia and Georgia, Tbilisi accused Moscow of an all-out cyber attack against the websites of nearly all government agencies and the country's leading banks.
Russia denied the allegations, but said that "individuals in Russia" might have been responsible.
Western cyber-analysts alleged that Russia's security services had likely played a key role in organising those attacks.
The United States Cyber Consequences Unit said the 2008 attacks highlighted the need for international cooperation on cyber security.
The two countries' brief but bloody conflict marked the culmination of the spiralling tensions over Georgia's bid to forge closer ties with the West, which has long angered Tbilisi's Soviet-era master Russia.
Saakashvili lives in a self-imposed exile in Ukraine after his second term as president ended in 2013.
The fervently pro-Western reformist former leader is wanted in Tbilisi on charges of abuse of power, which he denies.
A number of former top officials from Saakashvili's administration were jailed after his party lost in 2012 parliamentary elections to the current ruling party, Georgian Dream, headed by oligarch Bidzina Ivanishvili.
Opposition parties and Georgia's Western allies have denounced the prosecutions as a political witch hunt.
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