Hero Ouattara turned villain eight minutes from time when a yellow card for elbowing Ali Maaloul was changed to a red after the Botswana referee checked the touchline VAR monitor.
Burkina Faso held on to repeat 1998 and 2017 last-eight triumphs over Tunisia and advance to a semi-finals showdown with Senegal or Equatorial Guinea in Yaounde on Wednesday.
Success for the Stallions came five days after the landlocked west African country was plunged into political uncertainty when President Roch Marc Christian Kabore was ousted in a military coup.
Earlier, hosts Cameroon beat giant-killers Gambia 2-0 in Douala and await the winners between Egypt and Morocco on Sunday in the other semi-final.
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Burkina Faso started with captain and Aston Villa forward Bertrand Traore among 12 substitutes while an injury ruled out vice-captain and central defender Issoufou Dayo.
In the absence of the first and second-choice skippers, long-serving Belgium-based goalkeeper Herve Koffi led a country that came closest to glory in 2013 when finishing runners-up to Nigeria.
After being hard hit recently by a coronavirus outbreak, Tunisia were back to near full strength with regular captain Wahbi Khazri among those returning to the starting line-up.
Despite the presence of former Premier League forward Khazri, the Carthage Eagles were captained by veteran forward Youssef Msakni, who is appearing at a Cup of Nations tournament for the seventh time.
The Burkinabe Stallions kicked off boosted by the knowledge that they won all three previous Cup of Nations quarter-finals, including two against Tunisia.
Tunisia had been much less successful at the same stage of the flagship African tournament, winning just four of 10 last-eight matches.
The deadlock was broken three minutes into first-half added time with Dango Ouattara putting the Stallions ahead.
A pass from the centre circle found the midfielder from French Ligue 1 club Lorient on the right flank and he proved too quick for Oussama Haddadi before checking his run and firing into the net.
Tunisia coach Mondher Kebaier reacted to falling behind by taking off Haddadi at half-time and bringing on Ali Maaloul, who plays his club football with African giants Al Ahly of Egypt.
Saudi Arabia-based Naim Sliti was also introduced for the second half in place of Anis Ben Slimane on the right side of midfield.
As Tunisia pushed forward in pursuit of an equaliser they were exposed at the back early in the second half, but unmarked Blati Toure fired straight at Ben Said.
Maaloul is not only among the most polished left-backs in Africa, but also a set-piece expert and Koffi did well to push away a low free-kick from the Tunisian at the expense of a corner.
