Having been beaten by 281 runs in the series-opener at Centurion last week, the hosts slumped to 2-11 in Port Elizabeth on Thursday after winning the toss and choosing to bat.
However, Elgar - opening in place of Alviro Petersen despite being dropped from South Africa's list of centrally contracted players earlier this week - held firm with a composed 83, and received fine support from Faf du Plessis, who contributed 55 to a third-wicket stand of 112.
Nathan Lyon removed both set batsmen, but AB de Villiers struck 51 not out - setting a world record by managing a score of at least 50 for the 12th Test in succession - before bad light stopped play with 83 overs bowled.
Australia carried on where they had left off in the first Test by claiming two wickets inside the first six overs.
Graeme Smith was trapped lbw by Ryan Harris for nine, before Hashim Amla (0) departed in similar fashion to the in-form Mitchell Johnson.
Faced with a perilous situation, Elgar and Du Plessis fought hard to stabilise South Africa's innings, batting into the afternoon session.
The off-spin of Lyon eventually ended their partnership in the 44th over, as Du Plessis offered up a catch to Steve Smith at short-leg.
Elgar shared another useful stand of 58 with AB de Villiers before throwing his wicket away by skying Lyon to mid-off.
Smith got Quinton De Kock out soon after, with a full-toss that was heaved to mid-off, but De Villiers and JP Duminy made it through to the premature close.
De Villiers' stunning run of making at least a half-century in 12 successive Tests started against Australia in 2012 and the wicketkeeper-batsman is likely to be key to South Africa's hopes of pushing beyond 300 on day two.