Demoralized by a players strike two days before this first leg encounter in Port Harcourt, the Nigerian champions looked to have shaken off their troubles when Ifeanyi Egwim put them in front after just two minutes.
But the celebrations were cut short on 20 minutes when the visitors drew level.
Egwim had a chance to return Dolphins in front midway through the second half when he was brought down for a penalty, but he struck his kick straight at Coton Sport goalkeeper Fedjou.
Ghanaian Owusu Addae nicked a late winner two minutes to the end when he got on the end of Abduljeleel Ajagun's cross to jab home.
The result leaves Dolphins treading water as they head to Cameroon for the return leg in two weeks.
Nigeria's other Champions League representatives Sunshine Stars were handed a 4-1 shellacking by Angola's Recreative do Libolo.
It leaves the country in danger of having no representatives in the Champions League group stages.