Cuba last week hosted Russia's deputy prime minister Yury Borisov, who had previously visited Venezuela and Nicaragua, two other allies of Russia in Latin America.
Vyacheslav Volodin, the speaker of Russia's Duma or lower house of parliament, is to visit Havana later Wednesday.
Russia has amassed troops on the border with Ukraine, demanding guarantees that NATO will not expand its alliance eastward.
The Cuban statement said the United States -- which has maintained crippling sanctions against the communist island nation for six decades -- "has been threatening Russia for weeks" and "manipulating the international community" with an "anti-Russian propaganda campaign."
A month ago, the leaders of Russia and Cuba held telephone talks to boost their countries' "strategic association" and coordination on the international stage.
This came after Russian deputy foreign minister Sergei Ryabkov, in an interview in December, did not rule out Moscow sending forces to allies Venezuela or Cuba if diplomacy over Ukraine failed.
US National Security advisor Jake Sullivan has described the remarks as "bluster."
In Caracas, Venezuelan president Nicolas Maduro on Tuesday also expressed support for Russia.
"Venezuela announces its full support for President Vladimir Putin in the defense of peace in Russia, in the defense of the peace of the region," he said in a meeting with ministers broadcast on TV.
And Nicaragua's Daniel Ortega on Monday accused the United States and Europe of "using Ukraine to provoke Russia," which he said was merely "demanding security."
The United States and the then-Soviet Union are considered to have come closest to nuclear war in 1962 when Moscow deployed ballistic missiles to Cuba, setting off crisis diplomacy.
While Washington increased sanctions at the height of the coronavirus pandemic, prompting accusations of "cruelty" from Havana, Russia sent medical equipment and food.
On Tuesday, the Duma ratified a plan to restructure Cuban debt of $2.3 billion as the island nation battles its worst economic crisis in 27 years.