Australia's batsmen went through the campaign without scoring a century, the first time this has happened at home in a series of four or more Tests since 1882-83. Pacer Mohammed Shami also grabbed a couple of wickets while Jasprit Bumrah managed a wicket as well in the first innings.
Chinaman Kuldeep Yadav picked up five wickets for 99 runs.
Bad light and intermittent rain led to substantial loss of play with only 25.2 overs bowled on day four.
India's Kuldeep Yadav, second left, celebrates with teammates after taking the wicket of Australia's Nathan Lyon.
Mitchell Starc and Josh Hazlewood frustrated the Indian bowling line up adding 42 runs for the last wicket.
Yadav got his reward thereafter with Hazlewood trapped lbw on a close umpire's call, and the decision stayed in the wrist-spinner's favour despite DRS. On their home soil, this was the first instance of Australia following on since 1988, again in Sydney, versus England. It couldn't commence at the scheduled early 10 am local time start due to bad light and wet conditions from overnight heavy rain. Starc finished on 29 not out.
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Kuldeep claimed three wickets and Jadeja two as Australia collapsed from 122-1 at lunch to 198-6 shortly after tea in a pattern all too familiar for an Australian batting order yet to score a century this series.
India denied Australia any real possibility of a victory here by batting for almost two full days with Cheteshwar Pujara (193) and Rishabh Pant (159 not out) scoring hundreds.
With a series-levelling win out of the question, Australia will be hoping to hold on for a draw on day five of the final Test against India at the SCG.
Harris, who scored 70 in the second Test in Perth, brought up his 50 off 67 balls and looked set to be Australia's first centurion this summer until he was dismissed by Jadeja.
Australia 2nd Innings: Usman Khawaja batting 4; Marcus Harris batting 2.