Burkina Faso were beaten finalists in the 2013 AFCON, but bow out after picking up only one point from three games to finish bottom of Group A two years later.
Thievy Bifouma put Congo in front and substitute Fabrice N'Guessi Ondama grabbed a late winner soon after Aristide Bance had equalised.
Congo progress to the quarter-finals as group winners and hosts Equatorial Guinea also advance after beating Gabon 2-0.
A downbeat Burkina Faso coach Put was left to reflect on what might have been following his side's early exit.
He said: "It is difficult to explain. Luck was not on our side in all the three games. If you see the chances we have created in all the games and even today. We had three good chances but luck was not on our side.
"We were knocking on the door but did not enter. The two teams people were expecting to qualify from this group, Burkina Faso and Gabon, failed to qualify.
"Burkina Faso and Gabon has the same quality. There is still a long way to go."
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