Rain delayed the start of play for two-and-a-half hours and saw the match reduced to 42 overs a side, and the West Indies won the toss and sent South Africa in to bat.
Rossouw helped himself to a career-best 132 off just 98 balls, with nine fours and eight sixes, and Hashim Amla cruised to yet another century, scoring a fine 133 off 105 deliveries.
Rossouw and Amla added 247 for the third wicket off just 181 balls, a record third-wicket partnership for South Africa and equalling their own mark for the best stand for any wicket that they set in the second ODI at the Wanderers.
It was another top-class batting display by the South Africans, even with big gun AB de Villiers having a rest, as they laid a solid platform on a tricky pitch and then launched a spectacular late onslaught.
A couple of wickets had fallen in the first 10 overs as Quinton de Kock (4) and Faf du Plessis (16) fell to horizontal-bat strokes on a pitch that provided enough awkward bounce to make those shots difficult.
Amla once again looked in a class apart as he reached a 49-ball half-century in the 21st over, midway through the innings with South Africa on 109 for two, while Rossouw, who has endured a troubled ODI career thus far, struggled through to his 50 off 60 balls.
But those early miscues all seemed a lifetime ago as Rossouw then blossomed in spectacular fashion, needing just 23 more deliveries to register his second century of the series.
The left-hander raced past Amla and looked every bit a power-hitter as he smashed 82 runs off his last 38 balls.
Amla just kept on amassing runs as he has all series, except in this innings he was able to do it in half-dozens at a time, clearing the boundary six times to go with his 11 fours, showing that even a great batsman like him has had room to improve his game.
The West Indies were bleeding runs horrifically as the sun shone brightest on its way to the western horizon, bathing SuperSport Park in a sepia light befitting a display that was one for the ages.
The last 147 runs of the Amla and Rossouw partnership came off just 69 balls … and that by two batsmen who are not renowned for being big boundary-hitters.
Finishers David Miller (23 off 9) and JP Duminy (18* off 7) were left with only a handful of overs to bat but they collected their share of boundaries as 87 runs were hammered in the last five overs.
With the bounce providing something for the bowlers to work with – which the West Indians failed to do – South Africa’s total is certainly well over par. But they will be wary that Chris Gayle is probably due a score in this series.