Sudan, fielding a side made up entirely of home-based players against an Ivorian team comprising only players based abroad, did enough to deserve a point in the Group B game.
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Ivory Coast, with Drogba desperate to win something for his country at the age of 33, are among the favourites but will have to improve if they are to add to their only Nations Cup title won 20 years ago.
"I'm just very happy we got the points," Drogba told reporters. "We did some things well and some things not so well, but the main thing was that the win the match."
Sudan coach Mohamed Abdallah was proud of his side who are more than 100 places below their opponents in the FIFA rankings.
"My players lack experience, we have 16 new players in this team and playing their first international competition," he said.
"We had to take care not to give them too much space and not to make mistakes. I'm convinced the next match will be better. At least my players will now know what they have to do in the next match."
Gervinho and Jean-Jacques Gosso Gosso sent efforts high over the bar in a generally sluggish start by the Elephants, who seemed to struggle in the extremely muggy conditions.
The favourites broke through four minutes from half-time when Salomon Kalou provided a perfectly-flighted cross and Drogba headed in, possibly helped by a slight deflection.
However, instead of bringing the Ivorians to life, the goal seemed to have the opposite effect and they were lucky to reach the break with their lead intact.
Almost immediately after the goal, Mohamed Ahmed Basha shot wide from an excellent position for Sudan before Mudather Tayeb hit the crossbar in the next attack.
Undeterred, Sudan continued to push forward after half-time.
Tayeb's low cross nearly got Kolo Toure into a tangle as he just managed to clear his lines and, in the next attack, goalkeeper Boubacar Barry struggled to turn away Yousef Alaeldin's awkward bouncing shot.
Sudan were close again just before the hour when Tayeb had a header turned over by Barry.
Gervinho nearly added a second with a shot which squirmed through Mahjoub El Moez's arms but the Sudanese goalkeeper managed to grab the ball as it was about to creep over the line.
