Sydney Mohamed Sylla gave the Burkinabe a 1-0 half-time lead and closed the scoring after Daniel Etor had reduced arrears to offer the Nigerians hope.
Victory completed two joyous days for the Burkinabe, who reached the knockout stages via a drawing of lots after finishing level on points with Sudan in the group stage.
Ibrahima Keita scored twice as Senegal surprised hosts Congo Brazzaville 3-1 in the first semi-final.
Senegal took the initiative from the kick-off and Congo goalkeeper Christoffer Mafoumbi prevented a first-minute goal.
But the tall shot-stopper had no chance when an acrobatic overhead kick from Cheikhou Dieng looped into the net.
A defence-splitting pass and timely off-the-ball running by Keita set him up to fire past Mafoumbi for a 2-0 half-time lead. Congo improved dramatically after the break and Merveil Ndockyt reduced arrears through a close-range shot after a cross was not cleared.
Both sides had chances in an entertaining game before Keita put the outcome beyond doubt eight minutes from time with a header into the corner.
Sylla and Keita top the scoring charts with three goals each