Portuguese coach Paulo Duarte introduced giant striker Aristide Bance on 76 minutes as a tight, physical last-eight clash seemed destined for extra time.
Within five minutes Bance broke the deadlock in Libreville by slamming the ball into the net after a free-kick was pushed into his path just outside the box.
However, big-screen replays suggested the set-piece should not have been awarded with the ball hitting the chest of Mohamed Ali Yakoubi, not his right hand.
Prejuce Nakoulma doubled the lead on 84 minutes through a breakaway goal to clinch a semi-final against Egypt or Morocco at the same venue next Wednesday.
Success maintained a 100 percent Burkinabe record in Cup of Nations quarter-finals having beaten Tunisia (1998) and Togo (2013).
Since winning the title for the only time in 2004 when hosts, Tunisia have reached five quarter-finals, and lost them all.
“We produced high-level football,” said Duarte. “There was quality and confidence in our performance.
“The dream is to do better than in 2013 (runners-up to Nigeria), but it will not be easy.
“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.”
Tunisia coach Henryk Kasperczak said: “We did not show our qualities. We did not possess the rage to win.
“We were a little tired, the physical freshness was not perfect, we did not play quickly enough when attacking.”
The goalless first half produced more yellow cards than clearcut chances at the Stade de l’Amitie.
Tunisians Aymen Abdennour, Yakoubi and Syam Ben Youssef were cautioned by the South African referee, as was Burkinabe captain Charles Kabore.
Bertrand Traore, a Chelsea midfielder on loan to Dutch giants Ajax, came closest to scoring for Burkina Faso midway through the half.
Nakoulma did the spadework by muscling past two Tunisians and Traore turned smartly in a crowded goalmouth to fire just over.
Burkina Faso had a narrow escape on 34 minutes as Taha Yassine Khenissi flicked a corner across the goal and Yakoubi just failed to connect.
Soon after, Burkinabe Issoufou Dayo had the ball in the net after a free-kick, but the Tunisians were relieved to see the flag raised for a clear offside.
Having had less first-half possession, Burkina Faso reversed the trend as the second half unfolded, pushing forward without really threatening to score.
The fifth yellow card of the match went to Burkinabe Yacouba Coulibaly, the only local-based footballer in the starting line-up.
Team-mates Blati Toure and Nakoulma were also cautioned by the South African referee, bringing the match total to seven.
Most of the 10,015 spectators were backing Burkina Faso and they showed great excitement when Bance replaced Cyrille Bayala.
It proved an inspired substitution as the much-travelled crowd favourite ended the stalemate.
Bertrand Traore pushed a free-kick to the Ivory Coast-born striker and he hammered a low shot past Tunisia goalkeeper and captain Aymen Mathlouthi.
The lead was doubled when Nakoulma broke clear, raced past Mathlouthi, who had charged from his box, and stroked the ball into an empty net.