Drogba's header thumped into the back of the net from Chelsea teammate Salomon Kalou's cross in the 39th minute to cap a period of constant pressure.
Arsenal striker Gervinho had several chances to extend the lead but his finishing let him down, forcing Ivory Coast to endure a nervous finale.
Ranked more than 100 places below Ivory Coast, Sudan went closest when Mudathir El-Taieb's shot rattled the crossbar just before halftime.
Later, Angola plays Burkina Faso in the other Group B match at the Nuevo Estadio de Malabo.
Drogba led Ivory Coast's attack, with Yaya Toure positioned in an advanced midfield position to link moves and offer support.
Sudan coach Mohamed Abdallah opted for a single striker as his team aimed to provide an upset against Ivory Coast's highly paid superstars.
As expected, Sudan saw very little of the ball in the opening stages, with Kalou and Gervinho finding space to cause problems on the wings for Ivory Coast.
Gervinho hit a left-footed shot over the bar in the 12th minute and followed that with a header that was also too high on the half-hour mark.
The Arsenal striker wasted his third - and best - opportunity shortly after. Drogba brilliantly controlled a long ball floated deep into the box and laid the ball back, but Gervinho skied his shot well over.
The pressure finally told when Salomon Kalou checked inside his marker wide on the left and crossed into the box for Drogba to power a header into the net from close range.
Sudan might have been expected to crumble, but instead rallied and almost equalized.
Lone striker Ahmed Bashir had the first chance when he received the ball inside the box but pushed his shot wide on the turn in the 42nd.
Barely a minute later, Ivory Coast goalkeeper Boubacar Barry made a fingertip save to push El-Taieb's shot against the crossbar.
Barry was called into action again five minutes into the second half to push a long-range volley from Yousef Alaeldin away for a corner.
Gervinho's frustrating evening continued when he swapped passes with the influential Yaya Toure and burst into the box, but his shot was gathered at the second attempt by goalkeeper Mahjoub El-Moez.
Drogba should have given Ivory Coast breathing space, but he volleyed a cross from Igor Lolo wildly off target in the 68th.