Jos Buttler's 73-ball century helped the tourists to 399-9, with Ben Stokes, Alex Hales and Joe Root passing 50.
Quinton de Kock's unbeaten 138 off 96 balls kept the Proteas in contention.
But the loss of wickets at key times halted the home side's momentum before rain ended the game with them on 250-5 to give England a 1-0 series lead.
The second ODI in the five-match series is in Port Elizabeth on Saturday.
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Joe Root was the only batsman in England's top eight to score at less than a run per ball
The batting effort of captain Eoin Morgan's men represents a continuation of the dramatic improvement in England's limited-overs cricket since their woeful World Cup of 2015, when they were eliminated in the first round.
Since then, they have won a one-day series against World Cup runners-up New Zealand and Pakistan, as well as being narrowly beaten 3-2 by world champions Australia.
They have also won five consecutive Twenty20 internationals since the World Cup in an overall run of seven successive T20 wins.
