The 27-year-old left-hander smashed 11 fours and three sixes in what was only his third Test innings as an opener and first outside South Africa.
He also shared 70 runs for the opening wicket with Alviro Petersen (34) and another 124 runs with Faf du Plessis (57 not out) after new Test skipper Hashim Amla chose to bat on a placid track at the Galle International Stadium.
Elgar, playing his 10th Test, reached his second century in style, dancing down the track to smack off-spinner Dilruwan Perera for a massive six over the fence.
Elgar's form bodes well for a side going through transition after the retirement of Graeme Smith, Mark Boucher and Jacques Kallis.
Plessis, batting at number three for the first time in 15 Tests, also appeared at ease against the bowling attack, which relied heavily on spin.
Earlier, Petersen was particularly severe on left-arm spinner Rangana Herath, welcoming him with four boundaries in his first over of the day.
Elgar in turn picked fast bowler Shaminda Eranga for special treatment, hitting him for three fours in the same over, including one that crashed away through the point boundary.
Petersen was unhappy with the decision of on-field umpire Richard Kettleborough and asked for a review.
The decision was upheld when replays showed the ball would have clipped leg-stump.
South Africa have picked a lone specialist spinner in Imran Tahir with JP Duminy as a part-time off-spinner.
Sri Lanka chose two slow bowlers in Herath and Perera.
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