Speaking at a press briefing in Moscow, Peskov said the same "visible hand" is trying "to add anti-Russian elements" to the events in the capital Tbilisi.
"The situation that triggered these popular unrest has nothing to do with the Russian Federation.... At the same time we see that someone's hand is not 'invisible', it is quite visible.
"We can see where the president of Georgia, addressing (the Georgian) people ... from America. And someone's visible hand is diligently trying to add anti-Russian elements here," he said.
Peskov added that there are risks of destabilization in so-called Abkhazia and South Ossetia, partially-recognized states, bordering Georgia.
The spokesman also criticized Armenia's restrictions against some Russian citizens, amid recent Yerevan's ban on entry for several Russian lawmakers and media managers.
The official also called freezing of the Russian assets in the EU "a violation of international law and an encroachment on property," and also an "encroachment on the foundations of the world economy."
"It's illegitimate, it's illegal. We will fight against this," he added.
Commenting on remarks by Ukraine's Security Council Secretary Oleksiy Danilov, who called an "alarming trend" the increase of the number of Ukrainians, who speak in favor of peace talks with Russia, Peskov said: "Despite all the propaganda ... fooling of the Ukrainian population, there are still some trickles of sober thinking and approaches to the future of our bilateral relations in the country, and they are inevitable, because we are neighbors, this is obvious, and we have talked about this before."