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.
All the players involved were re-tested on Monday with only goalkeeper Ibrahim Sesay returning a negative result on Tuesday.
It meant that the Leone Stars had a depleted squad of 20 players, including two goalkeepers, for the game.
The Sierra Leone delegation complained that in the build up to Monday’s game they had been informed by a Confederation of African Football official that the whole squad were negative after tests carried out in Guinea.
The players had also returned negative tests in Sierra Leone on Saturday before they travelled to Guinea for the match, which had been moved as the stadium in Freetown was not up to standard to host a Nations Cup qualifier.
After meetings at the stadium on Monday the Confederation of African Football (Caf) agreed to postpone the match by a further 24 hours.
It was the second time the qualifier had been hit by disputes over players returning positive Covid-19 tests, after the same issue saw the game called off on 30 March when it was originally due to be played.
