Members of the International Olympic Committee (IOC) meeting in Buenos Aires chose the Japanese capital, which previously hosted the Games in 1964, over Istanbul, after Madrid was dramatically eliminated following a first-round tie with the Turkish city yesterday.
Crowds gathered in Tokyo broke into wild cheers, despite the result coming in the early hours of the morning nearly a dozen time zones away, television pictures showed.
He told delegates that the situation was "under control", adding: "It has never done or will do any damage to Tokyo."
Fukushima operator the Tokyo Electric Power Co also played down fears of a possible spread of radiation in a separate message.
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Concerns over Fukushima -- seriously damaged in the devastating 2011 earthquake and tsunami that killed more than 18,000 people -- had dogged the bid in the final days but Abe successfully allayed IOC members' fears.
Tokyo -- third in the vote for the 2016 Games won by Rio de Janeiro -- had been the frontrunners originally, as they promised a safe and financially secure Games in what they termed "uncertain times".
The decision means it will be the fourth time that Japan plays host to the Olympics, having also organised hosted winter Games in Nagano (1998) and Sapporo (1972).