Kodjia squandered several earlier chances before finally breaking the deadlock in the 64th minute of the Group D match at the Al Salam Stadium in the capital’s suburbs.
The goal was scored at the end of a swift move out of defence by the Ivorians.
It took just two passes to find the forward, who was goalside of the defence and had only the keeper to beat.
Kodjia had been in a similar position in the first half but instead of placing the ball, he blasted it to allow South Africa goalkeeper Ronwen Williams to come up with a brilliant close-range stop in the 32nd minute.
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The Aston Villa striker was also twice set up with goal scoring chances in the opening 11 minutes, but both times failed to get a touch.
The Ivorians also struck the woodwork just after the half hour mark as Nicolas Pepe curled a left-footed effort on the crossbar.
Pepe had another set piece chance two minutes from time but Williams made a finger-tip stop to deny the Ivorians a second.
With the temperature in the stadium touching 36 degrees Celsius, it stifled the tempo of what was a largely scrappy affair.
South Africa, who won their only Cup of Nations title in 1996, were unable to get their short passing game going and only had one half chance which captain Thulani Hlatshwayo headed wide of goal.
Two-time winners Ivory Coast join Morocco, who edged Namibia with a last gasp goal on Sunday, at the top of the group standings.
The Ivorians next meet Morocco on Friday while South Africa take on neighbours Namibia at the same venue.
(Reporting by Mark Gleeson; Editing by Pritha Sarkar)
