Ravindra Jadeja’s 4/62, followed by a fine maiden half-century from Shubman Gill as the visitors edged Day 2 of the third test at the Sydney Cricket Ground on Friday.
Resuming at overnight score of 166/2, the pair of Marnus Labuschagne and Steve Smith added 40 more runs before the former was caught at slip by Ajinkya Rahane off Jadeja for 91.
Matthew Wade in trying to attack the left arm spinner was caught at mid wicket by Bumrah.
Bumrah struck twice by dismissing Cameron Green leg before wicket for a duck and then accounting for skipper Tim Paine with a brilliant inswinger.
Neither Pat Cummins nor Nathan Lyon troubled the scorers as both fell cheaply to Jadeja.
Mitchell Starc made a useful 24 before top edging a Saini delivery to Gill.
Star batsman Smith was the last man to fall for a brilliantly made 131, run out by a superb throw from deep square leg by Jadeja.
Australia thus finished on 338, a decent score but far less then what they would have imagined after being 206/2 at one stage.
Ravi Jadeja was the pick of the bowlers with figures of 4/62, while debutant Saini and ace pacer Bumrah grabbed two apiece.
Trailing by 338 runs, India got off to a solid start with Rohit Sharma and Shubman Gill adding 70 runs for the opening wicket.
The former looked good in his knock of 26, before he chipped one straight back to Josh Hazlewood who completed a simple caught and bowled.
Gill looked in fine touch and made a composed half century, the first of his fledgling test career, before edging a Pat Cummins delivery to Green at gully.
The pair of Cheteshwar Pujara and Ajinkya Rahane kept the Australian bowlers at bay, as the visitors ended Day 2 at 96/2, 242 runs behind.
Stumps as the end of Day 2- India 96/2 (Shubman Gill 50, Rohit Sharma 26; Pat Cummins 1/19) Australia 338 all out (Steve Smith 131, Marnus Labuschagne 91. Ravindra Jadeja 4/62, Navdeep Saini 2/65, Bumrah 2/66)