Elgar carried his bat for much of the day to help the Proteas, who eased to an innings win in last week's first match of three, prosper in Port Elizabeth after they were asked to bat.
The left-hander's 121 steered South Africa to a score of 270 for two, with Faf du Plessis 99 not out at the close.
It appeared Denesh Ramdin's decision to bowl was inspired when Shannon Gabriel grabbed a crucial early breakthrough.
Alviro Petersen was the man to fall as, having jumped to reach a short-and-wide delivery, he proceeded to pick out Leon Johnson at cover to depart for 17.
However, Elgar and Du Plessis immediately showed the kind of application Petersen was lacking amid some threatening Windies bowling.
Du Plessis was particularly dogged, reaching the interval having managed just six off 35 deliveries.
If South Africa ending the first session just one down owed much to impressive batting, their remaining that way was thanks to a degree of assistance from the tourists.
Marlon Samuels and Devon Smith were each guilty of shelling Du Plessis early in the afternoon, while Ramdin failed to stump a charging Elgar off Sulieman Benn.
It was a sign of what was to come when Elgar, a ball after his let-off, charged down the pitch to hit Benn for a four that brought up his half-century.
He and Du Plessis, who was still looking unconvincing, required little good fortune thereafter as they accumulated expertly.
Elgar's effort brought a third format century, his innings ending when he edged debutant Kenroy Peters behind.
Yet South Africa did not skip a beat, Hashim Amla reaching 17 not out as Du Plessis ended one shy of three figures.