Needing a point to qualify, an acrobatic goal from Jonathan Kodjia after 37 minutes calmed home nerves.
But after Kei Kamara equalised midway through the second half, the tension was palpable.
Another away goal would have knocked the 2015 Nations Cup winners out but they held on despite a nervy finish.
Salomon Kalou was a surprise starter for the Elephants as he had been expected to miss out following the recent deaths of both his father and aunt.
