Inglis finished unbeaten on 85 from 111 balls, striking nine fours and two sixes, while Carey reached 51 from 126 deliveries with four fours as Australia negotiated the new ball after it was taken at 80.1 overs.
They added 44 runs in the final 10 overs to carry Australia through to the close without further damage.
The pair had come together with Australia 185/4 and initially absorbed the pressure, bringing up their 50 partnership from 108 balls.
Inglis then reached his half-century from 76 balls before the fifth-wicket stand passed 100 from 168 deliveries.
Carey followed him to fifty from 125 balls.
Inglis survived a South African review and the Australians stood firm to see off the closing overs.
The late partnership had become necessary after Jansen's intervention earlier in the day.
Australia had moved to 184/3 through Renshaw and Green's commanding 126-run stand before Jansen finally found the breakthrough.
Renshaw was caught behind for 74 from 139 balls, with 10 fours. The dismissal gave Jansen his 100th Test wicket.
The lanky leftarmer struck again soon afterwards when Green was caught by Simon Harmer for 65 off 110 balls, including eight fours and two sixes.
That followed a morning in which Harmer had given the hosts the perfect start.
Australia began the day on 27/0, but Travis Head fell for 41 from 49 balls when Harmer bowled him at 58.
Marnus Labuschagne then fell without scoring, caught by Aiden Markram off Harmer, leaving Australia 58/2.
