The qualifier had twice been postponed due to coronavirus-test issues.
Kamara converted from the spot after 19 minutes and the Leone Stars held on for a famous victory to join the other 23 teams at the finals in Cameroon in January.
Sierra Leone needed to beat Benin – who required just a point – to seal second place behind Group L winners Nigeria.
The tie was moved to Tuesday from Monday after Covid-19 issues saw the game delayed just ahead of kick-off.
Both teams were at the stadium ready for the fixture when Sierra Leone were informed that six of their players had returned positive Covid-19 tests.
Sierra Leone qualify for Africa Cup of Nations for the first time in history
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Jubilation sweeps Freetown and Conakry as Sierra Leoneans celebrate Leone Stars qualification for the Africa Nations Cup – Cocorioko
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