The Stallions will now await the winner between Nigeria and Tunisia for their quarterfinal fixture while the Panthers failed to make it to the last 8 for the third consecutive AFCON.
Both Burkina Faso and Gabon turned up the pressure in the early stages with the Panthers playing through the wings mostly and on the front foot.
The Stallions were content with playing the waiting game, looking to catch Gabon on the break with swift counterattacks.
One of such fast breaks pitted Burkina Faso's Issa Kabore against Gabon's Sidney Obissa on the right wing with Obissa giving away a penalty and earning a yellow card for a rough tackle just inside the box.
The Burkinabe's captain, Traore stepped up to take the 18th-minute penalty but the Aston Villa forward hit the crossbar with his effort.
Burkina Faso stuck to their game plan and repeatedly got behind the high backline held by Patrice Neveu's men, catching the Gabonese out of position a couple of times.
Traore made up for his costly miss earlier by finishing another swift counterattack in the 28th minute to give the Stallions the lead.
It was a feisty first half with tackles flying around and five players going into the referee's book.
Neveu made three changes early in the second half, bringing on Denis Bounga, Axel Meye, and Louis Autchanga for Jim Allevinah, Anthony Oyono, and Alex Moucketou-Moussounda in a bid to get more creativity and attacking impetus.
Gabon had a penalty shout waved off by the referee in the 62nd minute as the changes briefly swung the pendulum in their favour.
The match reached boiling point when Obissa was shown another yellow card in the 67th minute for pulling back Boureima Bande who was looking to get a return ball from Traore.
Denis Bouanga and Ecuele Manga also picked up yellow cards while remonstrating with the referee just before Issa Kabore of Burkina Faso was also booked.
Traore went close to doubling his tally in the 74th minute but Gabon's goalkeeper Jean Noel Amonome got down quick enough to deny the 26-year-old.
As the match wore on, Bouanga, Meye, and Kabore all got chances to secure the result for Kamau Malo's side but they failed to make it count.
The referee showed two more yellow cards to Mohamed Konate and Saidou Simpore of Burkina Faso to take the number of cards shown in normal time to 13.
Gabon made Burkina Faso pay for their inability to close off the match by scoring a 91st-minute equaliser with Ecuele Manga pressuring Adama Guira to put the ball into the back of his net.
Both teams continued to play high-octane football with sweeping attacks and swift counter-attacks from end to end in the first half of extra time.
Burkina Faso had the ball in the back of the net in the 100th minute through Abdoul Tapsoba but the referee was quick to rule it out for offside in the build-up.
Tapsoba also got a yellow card for pulling off his shirt in his short-lived celebration. Gabonese reserve goalkeeper Anthony Mfa Mezui also got a yellow card in the 104th minute despite being on the bench.
Amonome pulled off a raft of point-blank saves in second-half extra time to keep Neveu's men in the game as Burkina Faso attacked in numbers in a bid to make their numerical advantage count.
The Gabonese remained brave in defence and their occasional attack to force the game to penalties.
Gabon and Burkina Faso both scored their first three penalties respectively before Guelor Kanga and Saidou Simpore missed one apiece for their countries. Andre Poko and Steeve Yago both scored the next penalties to send the shootout to sudden death.
Four rounds of penalty kicks followed till Lloyd Palun missed for the Gabonese and Ismahila Ouédraogo scored the 14th kick to send the Burkinabes to their fourth AFCON quarterfinals.