The result secured them second place in Group C to join Madagascar and Mauritania as countries going to the finals for the first time, and their achievement sparked wild celebrations in the east African country.
They were joined later on Saturday by Cameroon, Guinea Bissau and Namibia, who qualified despite losing 4-1 to Zambia in dramatic circumstances.
While Burundi needed only a draw, visitors Gabon - who last hosted the finals in 2017 - had to win to join Mali in the expanded 24-team finals in Egypt.
More: https://theworldgame.sbs.com.au/burundi-make-history-with-first-cup-of-nations-qualification