The Elephants' emphatic 5-0 victory in the World Cup qualifiers was marred by the disaster at Felix Houphouet-Boigny stadium, where at least 19 home fans were killed and over 130 were injured.
"It's a national drama," Drogba, who scored twice, told French channel Infosport after the Group E tie.
"We only learned about it in the evening after the match. We were shocked, stunned, when we found out there were deaths. We had difficulty understanding what had happened.
"These are the kinds of things that are harming the development of football in Africa."
Panic
Ivorian Football Federation president Jacques Anouma lamented the tragedy, which is being discussed by the nation's Prime Minister in the meantime.
"The situation escalated as panic set in with nearly 50,000 fans in and around the 35,000 capacity stadium," Anouma told local newspaper Le Patriote.
"We can only deplore what happened. We are obliged to wait until the end of the inquiries that the Minister of Security himself has ordered so that we know what happened."
"We really can't understand what happened at a match where everyone was warned it was a sell-out. We are really sorry. We give our condolences to the families of the victims and we wish a speedy recovery to the injured.
"We have played more important matches, things have happened, but not like this. We can't understand it. What I ask of Ivorians is not to start useless arguments, but to ensure that these kind of things don't happen again. It's sad. I'm deeply upset."
Ivory Coast's next home game will be against Burkina Faso in September.