US President Barack Obama also called for the trade embargo to be lifted.
He said that he was confident that the US Congress would do so soon.
The two leaders are expected to meet later on Tuesday in New York.
Speaking at the UN General Assembly on Monday, President Obama said he thought the Republican-held Congress would inevitably lift "an embargo that should not be in place anymore" despite its reservations over the administration`s support for it.
He said that while the Cuba policy of the US had "failed to improve the lives of the Cuban people", human rights remained a concern in relations with Havana.
His words evoked a positive applause from the 193-nation UN General Assembly.
The embargo has been in place since 1960 and remains a contentious issue in relations between Cuba and the US.
President Castro for his part said that now that diplomatic ties were back in place, the overall normalisation of relations "will only be achieved with the end of the economic, commercial and financial blockade against Cuba".
It was President Castro`s first address to the UN since succeeding his brother Fidel in 2006. Like President Obama, he received sustained applause.
The White House announced on Sunday that President Obama would hold talks with his Cuban counterpart on Tuesday on the sidelines of the UN General Assembly.
It will be their second meeting following on from their first historic get-together in Panama in April.
The UN General Assembly is set to discuss a new draft resolution criticising the US embargo at a meeting next month.
The assembly has voted every year since 1982 in support of a resolution calling on the US to end the embargo.
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