Ivory Coast striker Didier Drogba scored with a powerful header in the 39th minute, but Sudan refused to be cowed and gave the pre-tournament favorite some nervous moments in the second half.
Angola's victory at the Nuevo Estadio de Malabo came after Mateus Da Costa scored the opener in the 48th minute and former Manchester United forward Manucho hit the second in the 68th. In between, Alain Traore curled in a free kick for Burkina Faso.
Matches continue Monday, with co-host Gabon playing Niger and north African rivals Morocco and Tunisia clashing in Group C.
It's been 20 years since Ivory Coast's first African Cup triumph, and many believe this tournament is the nation's best chance of repeating the feat.
With traditional regional heavyweights like Egypt, Nigeria and Cameroon failing to qualify, Ivory Coast's highly paid, European-based stars are the standout side.
Against Sudan, the second-lowest ranked team in the tournament, Ivory Coast dominated possession without ever hitting top gear.
Gervinho was the main culprit for making his team's evening more worrisome than necessary, missing with a glut of scoring chances.
The Arsenal forward had already shot too high and headed over by the time Drogba gave him a chance to redeem himself in the 37th minute, but the shot ballooned off-target.
Drogba finally put Ivory Coast ahead with a solid header from Chelsea teammate Salomon Kalou's cross two minutes later.
Sudan rallied and could have scored a shock equalizer just before halftime, but Ivory Coast goalkeeper Boubacar Barry's fingertip ensured Mudathir El-Taieb's shot struck the crossbar.
That turned out to be the closest Sudan came to punishing Ivory Coast for its missed opportunities.
Ivory Coast continued to rack up chances in the second half - Gervinho's shot was gathered at the second attempt by goalkeeper Mahjoub El-Moez, substitute Max Gradel dragged a shot wide and Drogba volleyed over with the goal gaping.
Considering the surprises on the opening day of the tournament, when Zambia beat Senegal 2-1 and co-host Equatorial Guinea had a stunning 1-0 win against Libya, Africa's top-ranked team will be happy to have avoided conceding an equalizer.
Angola moved into prime position alongside Ivory Coast atop the Group B standings after a 2-1 win against Burkina Faso.
Both of Angola's goals were the product of defensive lapses.
An indecisive clearance by Burkina Faso's Bakary Kone presented Da Costa with the chance to open the scoring just after halftime, and Djakaridja Kone gave the ball away to allow Manucho to thump the winning goal in the 68th.
Alain Traore curled a free kick over the wall and into the bottom corner to equalize between those strikes.
