Paceman Mitchell Starc, who would be expected by Australia to play a role similar to that of Umesh with the ball, smashed an unbeaten 57 to frustrate the hosts and take his team past 250.
Earlier, David Warner made 38 and added 82 for the first wicket with Matt Renshaw after captain Steve Smith won the toss and opted to bat against a spin-heavy India.
Smith said he expected turn from the very first ball on the dry pitch at the Maharashtra Cricket Association Stadium in the Western Indian city of Pune, a venue making its debut as a test centre.
India captain Virat Kohli appeared to agree with his counterpart, bringing on spin spearhead Ravichandran Ashwin, the top-ranked test bowler, in just the second over of the day.
After four overs from paceman Ishant Sharma, Kohli opted to utilise spin from both ends when he brought on Jayant Yadav against the left-handed duo.
The sparse crowd thought Jayant had bowled Warner around his legs for 20, but the off-spinner had overstepped the crease and it was called a no-ball.
Warner and Renshaw were able to put loose balls away before Kohli brought on his other paceman, Umesh, in the 28th over.
Umesh needed just two deliveries to break the opening stand when Warner dragged the ball onto his stumps.
Australia then took an ultra-defensive approach against a side unbeaten in their last 19 tests, but more chances eventually came India’s way.
Smith and Shaun Marsh added 37 for their stand in almost 20 overs before the latter was caught down the leg side attempting to sweep Jayant.
The hosts struck two quick blows when left-arm spinner Ravindra Jadeja dismissed Peter Handscomb (22) leg before after the right-handed batsman had added 30 runs with Smith.
In the next over, Ashwin lured Smith into a drive taken by Kohli at midwicket to dismiss the world’s top-ranked test batsman for 27.
All-rounder Mitchell Marsh did not last long either, out leg before to Jadeja for four.
Renshaw, who had to leave the field in the morning with an upset stomach, returned to the crease to hit 68 before he was caught out at slip off Ashwin.
Umesh dismissed Matthew Wade for eight before taking the wickets of Steve O’Keefe and Nathan Lyon in successive deliveries.
Australia were boosted by Starc late on as the left-hander smashed five fours and three sixes to add 51 for the unbroken final wicket with number 11 batsman Josh Hazlewood, not out on one.
(Editing by John O’Brien and Hugh Lawson)