Elgar and captain Hashim Amla (38 not out) saw South Africa home as they reached 124-2 wickets before lunch on the final day to secure victory.
"It is good to still be the number one Test team but it's not really about the ranking,” said Amla. “It needs to be about being a winning team for a long time.”
They were 9-1 overnight and showed no hurry to knock off the remaining 115 runs needed for victory.
The first 44 balls of the day failed to deliver any runs before Elgar finally stroked a boundary off Jerome Taylor to break the stranglehold.
Faf du Plessis was out for 14 just after the first hour, handing Sulieman Benn his second wicket of the innings after the spinner bowled opener Aviro Petersen with the last ball of day four on Monday. The tall left-arm spinner toiled for 16 successive overs on Tuesday to finish with figures of 2-24.
Yesterday’s win added to South Africa's victory by an innings and 220 runs in the first Test in Pretoria. The second Test in Port Elizabeth was a rain-affected draw. Rain also threatened to play a role in the final Test as half of the fourth day was lost to bad weather, offering the West Indies a chance to create a contest after clawing back from a 92-run first innings deficit.
But their resistance crumbled after Marlon Samuels lost his wicket at 182-4 in the second innings, out for 74 to an injudicious slog that saw him caught on the boundary. It precipitated a collapse as their next six wickets fell for a paltry 33 runs as they were dismissed for 215, leaving South Africa a modest target of 124 to win.
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