Two goals in the closing 10 minutes handed Tunisia a fifth quarterfinal defeat in the competition since they were crowned champions back in 2004.
A tense, goalless game looked to be drifting towards extra time until substitute Aristide Bancé scored from a free kick in the 81st minute.
Bance, who had replaced Cyrille Bayala moments earlier, struck a thumping low drive past Tunisia's goalkeeper Aymen Mathlouthi after Syam Ben Youssef had been penalized for handball on the edge of the penalty area.
Three minutes later the 2013 runners up sealed victory as Préjuce Nakoulma broke away down the right wing, rounded Mathlouthi -- who had advanced out of the penalty area -- before calmly slotting home into an empty net.
"We produced high-level football," Burkina Faso manager Paulo Duarte told reporters after the match.
"There was quality and confidence in our performance. The dream is to do better than in 2013, but it will not be easy," Duarte added referring to the Stallions loss in the final that year.
"Do not forget that Tunisia had a day less to rest (after the group stage) than us. I thought they were sturdy at the back, but a bit slow."
Burkina Faso will play either Egypt or Morocco in the semifinal on February 1.
Another tense affair was played out in Franceville where Cameroon scraped through against Senegal following a penalty shootout.
Neither team could find the back of the net during 120 minutes of open play and so the game went to penalties.
After eight flawless spot kicks, the game swung on the boot of Sadio Mane whose weak strike was met by the outstretched arm of Cameroon keeper Fabrice Ondoa.
Vincent Aboubakar then blasted the ball high into the top left corner sending Cameroon's supporters wild and their team into last four where they will meet either DR Congo or Ghana.