The Stallions, runners-up in South Africa two years ago, dominated the first half but failed to find an opener, before the hosts improved after the break to secure their second point of Group A that left them second behind Gabon.
The visitors, meanwhile, claimed their first point of the campaign that still left them last behind Congo.
Most of the pressing in the first half came from the Burkinabe, who twice struck the left upright through the impressive Alain Traore.
The first came after 19 minutes when the Lorient striker saw his 20-yard free-kick rattle the left upright, while he again did the same in the 39th minute, only this time his volley from Djakaridja Kone's right-side cross was parried onto the same upright by Felipe Ovono.
It was the Red Devils' first win at the continental championship since 1974 and allowed them to climb to the top of Group A.
The match got off to an eventful start with Panthers forward Frederic Wagha forcing Christoffer Mafoumbi into a smart save with his effort after a deft Malick Evouna back heel.
As the half continued the contest became more cagey with Pierre-Emerick Aubameyang causing the Congo defence plenty of problems with his pace.
Jorge Costa's side had a few more half chances as the teams went into the break level pegging.