Mr Trump, who takes office in January, said he hoped Cubans could move towards a freer future.
Castro came to power in 1959 and ushered in a Communist revolution. He defied the US for decades, surviving many assassination plots.
Supporters said he returned Cuba to the people. Critics called him a dictator.
His brother Raul, who succeeded him as president, announced his death on state television on Friday night.
In a statement, Mr 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 Barack Obama, the relationship warmed and diplomatic ties were restored in 2015.
Mr Trump roundly criticised Mr 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".
Mr Obama, meanwhile, said history would "record and judge the enormous impact" of Castro. America was extending "a hand of friendship to the Cuban people" at this time, he added.
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