Coach Mazda’s drive via
Watani club, scored the only goal of the two-leg tie just after half-time in Al Obeid.
Burundi, hoping to eliminate Sudan again after beating them on penalties in 2014, paid for failing to establish a first-leg lead in Bujumbura.
Sudan face Ethiopia this month with the winners qualifying for the 16-team event restricted to players playing in their country of birth.
The competition, whose games have international status, was the brainchild of former Caf president Issa Hayatou, who lost his job last March after 29 years in charge.
Concerned that Africans playing in Europe were dominating national squads, he devised the Chan to give home-based players international exposure.
Sudan seek a second appearance after hosting the tournament in 2011 and finishing third behind Tunisia and Angola.
Success against Burundi gave a needed boost to football in the east African state, whose only Caf title came 47 years ago when they won the Cup of Nations as hosts.
Sudan were briefly barred last month by Fifa from competing internationally over government interference.
The national team made a disastrous start to 2019 Cup of Nations qualifying in June, losing 3-1 to Madagascar at the same ground where they overcame Burundi.
Meanwhile, Morocco lost to Cameroon in the opening round of 2019 Afcon qualifying, and are also losing to them in the battle to host the tournament.
Delegate Faouzi Lekjaa of Morocco bragged before the recent Caf symposium in his country that the north African kingdom would replace Cameroon as 2019 hosts.