Madagascar advanced to four points from their opening two games and even if they do not finish in the top two places in the group, probably have enough points to fill one of the places in the last 16 for the four best third-placed teams.
After holding Guinea to a 2-2 draw in their opening game, the Malagasy went one better with their first ever finals victory.
Midfielder Ilaimaharitra blasted home in the 76th minute after Burundi had given away a free kick on the edge of their penalty area.
Madagascar made a lively start with Carolus Andriamatsinoro having two good first-half chances, the second one of which was cleared off the line by Burundi midfielder Gael Duhayindavyi.
But Burundi proved the stronger side after the break, ironically after they took off a struggling captain Saido Berahino.
Fiston Abdul Razak sent a header just wide in the 64th minute ands Burundi might have opened the scoring four minutes later but a free kick from Gael Bigirimana was saved.
Then Ilaimaharitra broke the deadlock and might have scored a second from a similarly placed free kick two minutes from the end but this time it was saved by Burundi goalkeeper Jonathan Nahimana.
It was the second loss for Burundi, also competing at the finals for the first timer, after a 1-0 defeat by Nigeria.
Nigeria top the group with six points while Guinea have one. Madagascar finish the group stage against Nigeria on Sunday while Burundi meet Guinea at the same time.
Kenya, helped by a Michael Olunga brace, twice came from behind to beat East African neighbours Tanzania 3-2 on Thursday and chalk up their second-ever win at the Africa Cup of Nations.
Simon Msuva and Mbwana Samatta put Tanzania in front but Olunga and Johanna Omolo levelled for the Harambee Stars before Olunga struck the winner nine minutes from time.
Kenya, who had won only once in 15 previous matches in the competition, have three points from two games in Group C, level with Senegal, who lost 0-1 to Algeria on the same day.
