Abedi Bigirimana opened the scoring in the first minute, while Bonfils-Caleb Bimenyimana and Hussein Shaban scored the other two goals in the 8th and 19th minutes respectively, giving the hosts three goals in the first 20 minutes.
Namibia's Peter Shalulile made it 3-1 just before the break, while substitute Wendell Rudath added the second with five minutes remaining.
With four points, Burundi is now third in the three-team Group, on level with second-placed Cameroon and one point behind Group leaders Namibia.
Burundi faces a tough challenge in the last round of the qualifiers as they visit Cameroon next September.
The Group was reduced to three teams after Kenya's disqualification by FIFA due to the suspension of the country's Football Federation.
The first two teams in each Group will reach the Africa Cup of Nations set to be played in Ivory Coast in January next year.
Ethiopia held to a goalless draw by Malawi in a Group D match on Wednesday, as both sides fail to qualify for the Africa Cup of Nations.
Ethiopia and Malawi are third and fourth in the group with four points each.
Egypt, who defeated Guinea 2-1 on Wednesday, secured their place in the Ivory Coast tournament, leading the group with 12 points.
Guinea finished second in the group with nine points and also qualified for the continental competition.
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