Donald Trump, who takes office in January 20, said he hoped Cubans could move towards a freer future.
Fidel Castro came to power in 1959 and ushered in a Communist revolution. He defied the US for decades, surviving many assassination plots.
His supporters said he returned Cuba to the people. Critics called him a dictator.
Raul Castro, who succeeded him as president, announced his death on state television on November 25.
In a statement, Donald Trump said that while Cuba remained “a totalitarian island, it is my hope that today marks a move away from the horrors endured for too long, and toward a future in which the wonderful Cuban people finally live in the freedom they so richly deserve”.
The US cut ties with Cuba in 1961 amid rising Cold War tensions and imposed a strict economic embargo which remains in place more than half a century on.
Under President Barack Obama, the relationship warmed and diplomatic ties were restored in 2015.
Donald Trump roundly criticized Barack Obama’s policy on the campaign trail but made no mention of his pledge to reverse it in his statement, saying his administration would do all it could to ensure Cubans could “begin their journey toward prosperity and liberty”.
Barack Obama, meanwhile, said history would “record and judge the enormous impact” of Fidel Castro. America was extending “a hand of friendship to the Cuban people” at this time, he added.
Fidel Castro was the longest serving non-royal leader of the 20th Century. He had been retired from political life for several years, after handing power to his brother in 2006 because of illness.
He will be cremated on November 26 at a private ceremony in Havana and a period of official mourning has been declared in Cuba until December 4, when his ashes will be laid to rest in the south-eastern city of Santiago.
In Miami, where there is a large Cuban community, there have been celebrations in some parts of the city, with people banging pots and cheering.
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