Shola Ameobi and Emmanuel Emenike also found the net as the Super Eagles continued a 10-year run on home soil without defeat
In a repeat of the 2013 Africa Cup of Nations final, the hosts dominated proceedings and could have won by a greater margin had their finishing not lacked accuracy at times.
Ideye opened the scoring early on with a good piece of opportunism following a defensive mistake, and doubled his tally just before the break with a coolly-taken finish.
Shola Ameobi netted his first international goal in the second half, and Emmanuel Emenike made it 4-0 just after the hour before Saidou Simpore got a late consolation for the visitors.
With both sides having qualified for the next stage of World Cup qualifying after topping their groups, only local pride was at stake - with Nigeria looking to extend their 49-year unbeaten record against Burkina Faso while maintaining their 10-year run on home soil without defeat.
The hosts began well but saw John Ogu and Ameobi fire good openings wide, before Ideye opened the scoring in the 12th minute as he took advantage of a mix-up between defender and goalkeeper to roll the ball into the back of the net.
The visitors rarely threatened the Nigeria goal and they fell further behind just before the break as Ideye doubled his tally.
The Dinamo Kiev striker beat three defenders to latch onto a through-ball from Ogu and calmly lifted the ball over Lare Diarra.
Ameobi finally got his goal early in the second half after squandering several earlier chances as he fired into the back of the net after excellent work from Ideye.
Emenike added a fourth in the 63rd minute, Ogenyi Onazi creating space for the striker allowing him to fire the ball beyond Diarra.
Burkina Faso were struggling to get into the game but did manage a late consolation through Simpore, the defender meeting an Issa Gouo cross with a simple tap-in at the back post.