The unfancied West Africans won with two stunning left-foot strikes from Alain Traore and goals from Djakaridja Kone and Jonathan Pitroipa.
They took all three points despite having goalkeeper Abdoulaye Soulama sent off for a careless handball outside his penalty area after an hour.
Burkina Faso have four points from two games, ahead of holders Zambia and Nigeria, who both have two following their earlier 1-1 draw. Ethiopia are bottom with one.
Traore opened the scoring after 34 minutes and hit an unstoppable second 16 minutes from time. Kone slid the ball in after a good move on 79 minutes and Pitroipa struck in stoppage time as Burkina Faso won a finals match for the first time since 1998 when they hosted the event.
The game was disrupted briefly late on when a man ran on to the pitch in his underpants before being intercepted by officials and bundled away.