For the first time since 1992, Congo have progressed to the last eight of the tournament after Claude Le Roy led them to the top of Group A with seven points.
A quiet opening 45 minutes saw Burkina Faso have the better of the chances, but they were punished early in the second half.
Thievy put his body on the line to meet Ferebory Dore's low cross, the striker sandwiched between Steeve Yago and goalkeeper Germain Sanou as he registered his second goal of the tournament.
Burkina Faso – beaten finalists in 2013 – introduced star names Aristide Bance and Bertrand Traore from the bench to try and reignite their campaign, and the former gave them hope with an 86th-minute strike.
However, Congo substitute Fabrice N'Guessi Ondama secured their win, after Sanou had misjudged a punch.
Victory sealed top spot in the group for Le Roy's side, with host nation Equatorial Guinea also progressing after a 2-0 win over Gabon.
Burkina Faso started confidently and wasted two chances in the opening 11 minutes to score their first goal of the tournament.
Jonathan Pitroipa – top scorer in the qualifiers – was first to find space as Prejuce Nakoulma's cross found him 12 yards from goal, but he was unable to direct his seventh-minute header on target.
Four minutes later, Burkina Faso had two corners in quick succession, with the latter seeing Mohamed Koffi fail to convert at the back post.
Congo struggled to find their fluency in the final third in the opening exchanges, but with 21 minutes played Thievy should have broken the deadlock.
The striker latched onto a Delvin N'Dinga throughball inside the penalty area and held off a defender’s challenge, only to scuff his shot wide of Sanou's goal.
Late pressure from Burkina Faso saw Pitroipa presented with another clear-cut chance in the penalty area, but the Al Jazira striker blazed his volley over to end a disappointing half.
Congo began the second period with new attacking threat and after Dore had wasted two clear-cut chances, Thievy broke the deadlock.
Dore was released behind the Burkina Faso defence and delivered a low cross into the six-yard box, where the Almeria striker converted from close range.
A 73rd-minute free-kick should have produced a second Congo goal as a well-worked routine found N'Dinga on the edge of the penalty area, but he shot straight at Sanou.
Bertrand Traore should have levelled in the closing minutes but he first dragged a shot wide, and then saw an 82nd-minute effort ruled out for offside.
Burkina Faso did get a goal as Bance, on as a late substitute, met Issiaka Ouedraogo's cross, but their fightback was quickly ended as Ondama took advantage of Sanou's missed clearance.
There was still time for Alain Tarore to hit the crossbar with a long-range free kick as Burkina Faso pressed, but they had not done enough to stave off their disappointing exit from the competition.
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