The win, sealed by a second goal late on from substitute Fabrice Ondama, eliminated the group favourites, who had been runners-up at the last finals in South Africa two years ago.
Congo finished top of Group A with seven points, followed by hosts Equatorial Guinea, who beat Gabon 2-0 in Bata.Burkina Faso had to win to have any chance of advancing but were on the back foot after Bifouma scored in the 51st minute.
The Congo striker, who also scored on the opening day, slid in to bundle home a square pass from Ferebory Dore.
Burkina Faso had a magical run through the 2013 finals but there was no resemblance of that sorcery after yet another limp performance. They had lost to Gabon and drawn 0-0 with Equatorial Guinea in their opening two group matches.
They finally scored their first goal of the tournament after more than four hours when Aristide Bance equalised to set up a potentially thrilling final four minutes only for a goalkeeping error to allow Ondama to restore Congo's lead within one minute.
Bance converted a square pass from Issiaka Ouedraogo in the 86th minute but any chance of a grandstand finish ended when Germain Sanou punched his clearance straight onto Ondama for Congo's winner.
Congo now play in Saturday's quarter-final in Bata against the runners up in Group B, which will be determined on Monday.
Congo last appeared at the Nations Cup finals 15 years ago and have only twice made it past the first round, in Cameroon in 1972, when they won the tournament, and in 1992, when they were knocked out in the quarter-finals.
(Reporting by Mark Gleeson; Editing by Ian Chadband)
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