Pierre-Emerick Aubameyang's Gabon went into the match in Bujumbura needing to win, as they began the day two points adrift in Group C, but they were unable to clinch the victory which would have booked their place in the tournament.
Stubborn Burundi did well defensively, before taking the lead in the 77th minute when Cedric Amissi found the net, a goal that initially had them going top.
An own goal from Omar Ngando soon after ensured a nervy finish, but Burundi held firm to claim an historic point.
Burundi will join Group C winners Mali at the tournament, which will be played in June and July this year.
For Gabon and Aubameyang, meanwhile, elimination from AFCON qualification marks their latest disappointment in international competition.
The nation could only manage group stage finishes in the last two editions of the tournament and missed out on the 2018 World Cup after finishing third in their group in the final round of qualifying.
Aubameyang managed just a single goal in this AFCON process, which saw Gabon crash out with two wins, two draws and two defeats.
The 2019 African Cup of Nations makes history as the first to be held in June rather than the traditional January/ February slot, and is also bigger than ever before.