The Italian coach, 53, has a two-year deal and his task is to take the 1938 and 1954 World Cup finalists to the Euro 2020 tournament, when Hungary will host four matches.
Leekens was appointed last October but failed to make an impact as Hungary lost three and drew one of their friendly games, with the HFA saying in a statement that he had brought little improvement during his spell in charge.
Rossi is more familiar with the Hungarian championship as he won the league with Budapest Honved in 2017 but left soon after and went on to manage Slovakia's Dunajska Streda.
Hungary qualified for Euro 2016, the first time they had reached the European Championship finals since 1972, but the country last made it to a World Cup tournament in 1986.
They finished third in qualifying Group B for this year's finals in Russia but were a massive 14 points adrift of European champions Portugal and Switzerland who reached the finals.
(Reporting by Zoltan Fazekas; Editing by Ken Ferris)
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