The 24-year-old, who also scored the winner in the 2-1 quarter-final win over Ivory Coast, struck after 40 minutes when the ball bounced off a defender and into his path as he advanced on goal.
Mba flicked it over defender Mohamed Koffi then reacted quickest to score with his other foot low into the corner of goalkeeper Daouda Diakite's net for a contender for goal of the tournament.
Underdogs Burkina Faso, in the final for the first time, seemed over-awed by the occasion, which was played out in front of a near 90,000 capacity crowd, until the last 20 minutes when they threw everything forward in search of an equaliser.
But Nigeria with John Obi Mikel superb in midfield and Efe Ambrose, Kenneth Omeruo and Elderson Echiejile dominant at the back always held the upper hand.
The victory meant Nigerian coach Stephen Keshi became only the second man to win the tournament as player and coach. It was a third African title for Nigeria and their first since Keshi captained them to victory in 1994.