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LIVE UPDATES: SA v Australia - Day 1
South Africa were 145 for three at tea in Port Elizabeth after losing captain Graeme Smith and Hashim Amla inside the first six overs.
Elgar (61 not out) and Du Plessis (55) put on 112 for the third wicket - only the second century stand against Australia since the start of their 5-0 Ashes sweep against England earlier in the summer.
The left-handed Elgar, opening the batting in place of the ill Alviro Petersen, saw Smith and Amla depart before he opened his score off the 20th ball he faced.
But he never lost his composure and handled the pace of first-Test destroyer Mitchell Johnson with courage and skill, albeit in conditions more batsman-friendly than in the first Test at Centurion.
Elgar had faced 146 balls by tea, hitting six fours and two sixes, with both sixes against off-spinner Nathan Lyon
Ryan Harris made the first breakthrough when Smith played across the line and was leg before wicket for nine.
Johnson followed up with his 50th Test wicket in seven matches since the start of the Ashes series against England when Amla was beaten for pace and trapped leg before with a full-pitched delivery.
Elgar and Du Plessis steadied the innings with solid but cautious batting as South Africa reached 64 for two at lunch.
They picked up the pace after lunch, adding 59 in an hour before Du Plessis was out off the first ball of the afternoon drinks break, turning a ball from off-spinner Nathan Lyon straight into the hands of Steve Smith at short leg.
Du Plessis faced 125 balls and hit five fours and a six.
Elgar was one of three changes to the South African team which was beaten by 281 runs in the first Test at Centurion.
In a surprise move, left-handed batsman Quinton de Kock, who was not part of the original squad, was flown in and chosen bat at number seven to fill the place vacated by injured all-rounder Ryan McLaren.
Left-arm fast-medium bowler Wayne Parnell replaced left-arm spinner Robin Peterson.
There was almost a fourth change. The toss was slightly delayed while South Africa assessed the fitness of opening bowler Vernon Philander, who Smith said felt a twinge when he was warming up.
Australia were unchanged and almost immediately took over from where they had left off at Centurion, bowling straight and attacking the stumps in cool, overcast conditions on an easy-paced but well-grassed pitch.
Teams:South Africa: Graeme Smith (captain), Dean Elgar, Hashim Amla, Faf du Plessis, AB de Villiers, JP Duminy, Quinton de Kock, Wayne Parnell, Vernon Philander, Dale Steyn, Morne Morkel
Australia: Chris Rogers, David Warner, Alex Doolan, Michael Clarke (captain), Steve Smith, Shaun Marsh, Brad Haddin, Mitchell Johnson, Peter Siddle, Ryan Harris, Nathan Lyon
Special moment for Quinton de Kock who became the 82nd Proteas Test player since readmission this morning - CSA
