Leading the global praise, Pope Francis sent "warm congratulations" to US President Barack Obama and Cuban President Raul Castro for overcoming "the difficulties which have marked their recent history".
The announcement of the deal and the release of prisoners from both countries followed more than a year of secret talks in Canada and at the Vatican, directly involving the Pope, the first Latin American pontiff.
Obama discussed Cuba with the Pope during his visit to the Vatican in March and continued to work with the Holy See thereafter.
President Obama yesterday said the "rigid and outdated policy" of isolating Cuba since then had clearly failed and that it was time for a new approach.
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Castro, meanwhile, has urged the US to ends its trade embargo, which has been in place since the Cuba turned to Communism more than 50 years ago.
The European Union, which is in the process of normalising ties with Cuba, described the US-Cuba move as a "historical turning point".
"Today another Wall has started to fall," said EU foreign affairs head Federica Mogherini, adding that the 28-member bloc hoped ultimately to be able to "expand relations with all parts of Cuban society".
"This news is very positive. As much of the membership of the UN has repeatedly emphasised, it is time that Cuba and the US normalise bilateral relations," Ban told reporters in his end of year press conference.
The United Nations General Assembly has been tabling a resolution for over 20 years calling for an end to the US' economic, commercial and financial embargo on Cuba.
This year India also was among 188 nations that voted in favour of the resolution, which for the 23rd year in a row called for an end to the embargo.