Zambia had gone into the first leg trailing 1-0 but the Burundi side steamrolled over Zambia via a 3-0 result.
The African champions will miss out on the Niger party but will rue missed opportunities at home.
Zambia won the Under-20 crown on home soil in 2013.
Meanwhile the Under-17 national team beat Lesotho 1-0 at the ongoing Cosafa Castle Cup in Mauritius to amass four points and set a crunch encounter against South Africa on Tuesday.
Benson Simwanda scored an 89th minute goal to dump the Lesotho side out of the competition but will play the final formality match against Mozambique on Tuesday.