The South African captain suffered what was described as a soft tissue injury to his left ring finger during fielding practice on Thursday.
He had X-rays which revealed no break but it was decided after a visit to a hand specialist that he should not play.
Vice-captain Jacques Kallis will captain South Africa for the first time in the match starting on Friday, while Boeta Dippenaar will open the batting in Smith's place.
His withdrawal means Smith will not have the chance to finish his Test battles with Australia on a high note after a summer of failure.
In five Tests for South Africa he had a highest score of 40, while he made 12 and 0 when he captained the World XI against Australia earlier in the season.
In six matches against the world champions he made 242 runs at an average of 20.16.
The rest of the South African team was not named. Despite Smith's absence, the selectors have 15 players to choose from following the addition of batsman Hashim Amla and fast bowler Dale Steyn to the squad following South Africa's 112-run defeat in the second Test in Durban Tuesday.
The win clinched the series for Australia. It was also their fourth successive win against South Africa after they won a three-Test series 2-0 in Australia before the current tour.