Two thumping left-foot shots from Alain Traore were followed by late goals for Djakaridja Kone and Jonathan Pitroipa as Burkina Faso recorded their first win at the tournament since 1998.
They took all three points despite having goalkeeper Abdoulaye Soulama sent off for a careless handball outside his penalty area after an hour.
The Burkinabe, one of the outsiders in the group, now need only draw in their last game against holders Zambia at the Mbombela Stadium on Tuesday to progress to the knockout phase for the first time since they hosted the tournament 15 years ago.
"It was a bit of bizarre game, we were under pressure for the first 15 minutes but we showed discipline and attacked with great success," said Burkina's Belgian coach Paul Put.
"My team were grand tonight. It's a historic night for us. This is great for the confidence and I think we are on the right track. We have everything in own hands now against Zambia. But we need to keep both feet on the floor."
For Ethiopia, who hit the post in the fourth minute, defeat was a reality check on their return to the finals for the first time in 31 years after they had held Zambia to a 1-1 draw on Monday.
Traore's two goals in the 34th and 74th minute were both scorchers that emphasised the superiority of Burkina Faso, strengthened by the return from suspension of inspirational captain Charles Kabore.
Kone added the third in the 79th minute after winning the ball in midfield, playing a one-two with Pitroipa and finishing with a strong surge at goal.
Pitroipa's effort came deep in stoppage time and was also a result of a turnover in midfield as he was fed by Wilfred Balima and finished from close range.
Soulama's sending off capped an eccentric performance from the goalkeeper who had twice been caught trying to dribble the ball out of his goal but escaped being punished by Ethiopia.
He then caught a ball outside the area and was dismissed.
