Fast bowler Edwards took 2-11 as South Africa, who need 389 to win, stumbled to 45-3 at lunch on the fourth day.
South Africa have won eight of the nine Tests the teams have contested in the country and the other was drawn.
West Indies resumed their second innings on 146-8 and they were dismissed for 175 in the sixth over of the day's play.
Left-arm spinner Paul Harris took both wickets to finish with 4-35.
South Africa crashed to 20-3 in the first 10 overs before Jacques Kallis and Ashwell Prince stabilised the innings with a stand of 25 for the unbroken fourth wicket.
Fast bowler Daren Powell started the slide when he trapped Herschelle Gibbs in front without scoring in the third over.
Edwards first struck in the eighth over when he had Hashim Amla caught behind by wicketkeeper Denesh Ramdin for eight.
In Edwards's next over, Graeme Smith spliced a short delivery upward for short leg fielder Daren Ganga to dive across the pitch and complete the catch to dismiss the South African captain for 11.
Kallis was 16 not out at lunch with Prince on 10.
Scorecard:
West Indies first innings 408
South Africa first innings 195
West Indies second innings 175
South Africa second innings:
H Gibbs lbw b Powell 0
G Smith c Ganga b Edwards 11
H Amla c Ramdin b Edwards 8
J Kallis not out 16
A Prince not out 10
Total (for three wickets, 18 overs) 45
To bat: A de Villiers, M Boucher, P Harris, A Nel, D Steyn, M Ntini.
Fall of wickets: 1-4 2-17 3-20
Bowling: Powell 6-2-17-1, Edwards 6-1-11-2, Taylor 3-0-15-0, Bravo 3-2-2-0.
(Reporting by Telford Vice; Editing by Clare Fallon)
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