Substitute Aristide Bance scored with a low drive from a free-kick only five minutes after coming off the bench.
Prejuce Nakoulma made sure of the victory on the counter-attack, after Tunisia had thrown everyone forward, rounding the goalkeeper 40 yards out before slotting into an empty net.
The Stallions will face either Morocco or Egypt in the last four on Wednesday.
It is now three wins from their three quarter-finals appearances for Burkina Faso, who last reached this stage in 2013 when they finished as losing finalists.
For the first time they secured a quarter-final win in normal time although until Bance’s intervention, which finally broke Tunisia’s resistance, it appeared as though extra time was to be required again.
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