Emmanuel Amuneke’s men dominated the game from the very beginning, missing several chances.
Samuel Chukwueze came close in the 25th minute with a looping header while skipper Kelechi Nwakali fired narrowly wide from outside the box five minutes later.
Burundi got a big chance in the 17th minute but goalkeeper Akpan Udoh saved Nigeria.
Nigeria got the winning goal in the 68th minute through Chukwudi Agor who made a dummy before unleashing a powerful strike from the edge of the box that sailed into the net.
The return leg will be played on the weekend of June 10-12 in Calabar.
Victory for Nigeria in the return fixture will mean the Flying Eagles will take on Sudan in the third and final leg of the qualifiers in July.