The Stallions caused the second upset on Monday after Malawi crushed World Cup qualifiers Algeria 3-0 in Group A earlier in the day.
Ghana were scheduled to face Togo in the second Group B match but the latter have flown home after their bus was attacked by gunmen on Friday.
Ivory Coast created few goalscoring opportunities in the first half after finding it difficult to break down a resolute Burkina Faso defense.
Burkina could not make any attempts to hit Ivory Coast on the break but their tight defense annoyed the Ivorians.
Ivory Coast were harshly denied two penalties in quick succession when Emmanuel Eboue and Didier Drogba were brought down and tugged inside the area respectively.
They could have broken the deadlock on two occasions before the break.
Right-back Eboue saw a fierce shot from a tight angle goes narrowly wide on 29 minutes and Burkina keeper Daouda Diakite, who plays for Egyptian side Arab Contractors, produced a superb reflex save to tip over a close-range shot from Bakari Kone late in the first half.
Struggle
Ivory Coast found a way through Burkina's backline few times before halftime but they struggled to replicate such penetrations in the second period.
Despite the abundance of attacking options in the team's squad, they lacked creativity and imagination to score the decisive goal.
Their sole clear-cut chance fell to the disappointing Kone, who could not hit the target from six yards out.
In the 68th minute, Gervinho skipped past Paul Koulibaly on the left hand side of the area before cutting the ball back for the Marseille man, who shot wide with the goal at his mercy.
Kone was replaced with Abdelkader Keita as Ivory Coast coach Vahid Halilhodzic sought to add impetus.
Chelsea forward Salomon Kalou was also introduced but couldn't make a notable impact in the remaining minutes.