Badra Ali Sangare went to catch a ball headed back to him by a teammate in the third added minute in an attempt to prevent a corner, only to drop it into the path of Steven Caulker who squared for substitute Kamara to score.
It was a remarkable moment and one that prevented the Ivory Coast from wrapping up qualification for the last 16 there and then.
Sangare, who has taken over the gloves at this tournament because first-choice Sylvain Gbohouo was handed a doping ban, then went off injured, leaving right-back Serge Aurier to see out the match between the posts.
Before the late drama the 2015 champions appeared to have a second win in as many games in Group E sewn up thanks to Nicolas Pepe's second-half strike, but this result leaves them on four points and Sierra Leone on two.
"Honestly I can't explain that goal. You could have Jurgen Klopp or Pep Guardiola in charge and they wouldn't be able to prepare for that," said Ivory Coast coach Patrice Beaumelle, whose side must still face Algeria.