Barry, who appeared to cramp up during the shootout, denied fellow keeper Brimah Razak - one of five efforts missed - before slotting in the winner.
Victory gave the Ivorians their second title, in a repeat of the 1992 final.
That year, Ivory Coast won 11-10 on penalties also after a goalless draw.
They will be delighted to have emulated the feat having gone close twice since - in 2006 and 2012 - on the latter occasion losing to Zambia, who were then coached by current Ivory Coast coach Herve Renard.