USA officials said the decision relied heavily on Havana's ties to Venezuelan socialist President Nicolas Maduro, whom the Trump administration does not recognize as his country's legitimate leader.
The Trump administration move comes days before the inauguration of Biden, who has pledged to revive Obama's efforts at rapprochement between the two Cold War foes.
"The US political opportunism is recognized by those who are honestly concerned about the scourge of terrorism and its victims", Rodríguez tweeted.
One of the reasons Pompeo cites for the designation is Cuba's support for Venezuelan President Nicola Maduro.
CUBA hit out at being listed as a "state sponsor of terrorism" by the Trump administration today, countering that the island is itself a victim of terrorism financed and perpetrated by the U.S. government.
Meanwhile, the National Assembly of the People's Power of Cuba recalled the terrorist actions plotted from the United States against Cuba.
The move will penalize US companies for making business deals in Cuba and also restricts USA foreign assistance, bans defense exports and sales, and imposes controls on some exports, the State Department said.
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Cuba and the US, enemies after the late Fidel Castro took power in 1959, established diplomatic relations in 2015 when Obama was president and Biden was vice president.
Numerous voices in the region and the world also condemned the decision against Cuba, passed during President Donald Trump's last week in office.
And reversing the designation could be time-consuming and politically onerous, Emilio Morales, an exiled Cuban economist and president of the Miami-based Havana Consulting Group, told The Associated Press.
Rep. Gregory W. Meeks, D-N.Y., chairman of the House foreign affairs committee, said he was "outraged" by the designation, and urged the Biden administration to reverse it when it gets into office. Washington later enforced a trade and economic embargo on Havana.
Trump has reversed many of Obama's overtures to Cuba and also imposed sanctions on its leftist ally Venezuela, a stance that helped win support among immigrant communities in Florida, a crucial state in USA elections. "We believe that the USA should develop normal state-to-state relations with Cuba on the basis of equality and mutual respect".
Sen. Patrick Leahy, Vermont Democrat, called the move a "blatantly politicized designation [that] makes a mockery of what had been a credible, objective measure of a foreign government's active support for terrorism".
Blacklisting is expected to further hamper foreign investment in Cuba.
The terror designation severely hampers investment by foreigners who will now risk USA prosecution and can only be removed after a formal review by the Biden administration, meaning it may be in force for months.