The left-hander provided the backbone of a dominant day for the hosts in Port Elizabeth, as they reached 270 for two at the close having been put into bat.
Opening the innings, Elgar made 121 from 239 balls, and joined in a second-wicket partnership of 179 with Faf du Plessis, who was 99 not out at stumps.
Elgar eventually fell caught behind off the bowling of Kenroy Peters, having earlier been offered a reprieve by wicketkeeper and Windies captain Denesh Ramdin, who failed to stump him off Sulieman Benn.
That charge was a rare risk from Elgar, who conceded afterwards that an over-aggressive approach was unlikely to garner success.
"At times there were a lot of dot balls bowled to us," he said. "I think they had three maiden overs at us at a stage.
"It's one of the chats we had in the build up to this Test, the patience aspect. Even with the ball in hand you have to be patient, sometimes you don't pick up wickets as often as you want to.
"It's something that I have been fighting with myself personally throughout the last few games. I tend to get to the 20s and give my wicket away.
"It's nice to contribute in the manner that has just happened.
"Luckily, you have a guy at the other end that you communicate with. Faf and I get along quite well when we have the bat in hand - don't get me wrong we also get on well off the field.
"We have batted a lot together over the last four or five years from South Africa A and in the Test side. We have had a few substantial partnerships together so we know what to say to each other at the right time.
"There is a lot of seam movement out there. I think if you bowl the ball in the right area with a bit more consistency the rewards will be there.
"The Windies had their opportunities to have us at least five down by the end of play today but luckily we are only two down with a relatively decent score."