Second-half substitute Benson Omala scored the winner in stoppage time to see Kenya through, despite having been a goal down at half-time.
Patrick Otieno had netted the equalizer in the 62nd minute, cancelling out Burundian captain Hamimu Hakizimana’s opening goal at the stroke of half-time.
The Rising Stars progressed in second place from Group B after winning the first match 5-0 against Zanzibar before drawing 2-2 with archrivals Tanzania in Matchday Two.
They wound up their group assignments with another big win, a 4-0 scoreline against Ethiopia.
Meanwhile, Tanzania have also reached the last four after beating hosts Uganda 4-2 in another quarter-final match.