A sudden-death penalty from Ivory Coast goalkeeper Boubacar Barry last month denied a first continental title since 1982 to Avram Grant's men, who progressed to the last act despite a 2-1 defeat to Senegal in the group stage.
And Ghana failed once again to get the better of a Senegal team now under the leadership of new coach Aliou Cisse.
Both sides had chances to go in front in an even first half, but it was Senegal's Demba Ba who came closest when he hit the post from a promising position.
The match looked to have lost its spark at the beginning of the second period but a moment of magic from Konate earned Senegal a 66th-minute lead.
And the substitute - who had only been on the pitch five minutes when he opened the scoring - wrapped up the win with his second 10 minutes later, before Richmond Boakye headed a late consolation.
Asamoah Gyan was excused from international duty as he continues his recovery from an injury, and replacement Jordan Ayew dragged wide of the target when he got the first sight of goal after six minutes.
Senegal should have gone ahead soon after, with Ba slotting against the base of the left-hand upright having been sent one-on-one with goalkeeper Brimah Razak, before Wakaso Mubarak drew an excellent save from Abdoulaye Diallo at the other end.
Kwesi Appiah saw appeals for a penalty turned down as he went down just inside the box after 19 minutes, and Ghana looked destined to go ahead when the ball fell to John Boye in a dangerous area, only for the defender drill wide of the right-hand post.
Ghana were dealt a blow when Jonathan Mensah limped from the field after taking a knock 10 minutes before half-time and, with both sides playing attacking football, it was a wonder the tie remained goalless at the interval.
A series of substitutions contributed to the stop-start nature of the second period, with chances few and far between for both sides.
However, it was to be a spectacular strike that eventually broke the deadlock, with Konate – on as a replacement for Ba - controlling a ball from deep on his chest before looping a volley over a helpless Brimah from the edge of the box.
Konate was at it again 10 minutes later when he lifted over Brimah having been teed up by Sadio Mane.
Boakye pulled one back with a header six minutes from time, but it was too little too late as Grant was left with much to ponder ahead of Tuesday's friendly clash with Mali.
