Gernot Rohr's side, who are 89 places above Burundi in FIFA's world rankings, struggled for long periods in their Group B opener at the Alexandria Stadium but sealed all three points in the 77th minute.
Odion Ighalo, introduced for Paul Onuachu just four minutes earlier, latched onto Ola Aina's wonderful back-heeled pass and coolly whipped the ball past Jonathan Nahimana.
Nigeria will hope to put in a vastly improved performance against Guinea on Wednesday, while Burundi face Madagascar a day later.
At the other end, Samuel Chukwueze blazed over from Alex Iwobi's pull-back shortly before the half-hour mark, while an unmarked John Obi Mikel headed wide from six yards soon after.
Burundi ended the half strongly, with only a fingertip save from Akpeyi keeping out Frederic Nsabiyumva's looping header.
The stage was then set for Ighalo, the Shanghai Shenhua forward clinically lifting past Nahimana and in off the right-hand post to get his side up and running in the tournament.