At the close of play, the hosts were steadily placed at 121 for three in their second innings at the M A Chidambaram Stadium.
Back in the mix after 18 months, the 28-year-old left-arm spinner Ojha, bowling with a remodelled action, picked up five for 85, while leg spinner Amit Mishra managed three for 55.
There was a wicket, too, for medium pacer Abhimanyu Mithun.
In reply, India A were served in their second essay by Abhinav Mukund (40) and skipper Cheteshwar Pujara (42). Mukund and Pujara hit four and five boundaries each, respectively, before getting out towards the fag end of the day's play.
Both the wickets were taken by fast medium bowler Sandhu (2/13), who plays for Delhi Daredevils in the Indian Premier League (IPL).
Opener Lokesh Rahul had got out to Travis Head after making a 45-ball 29. His knock was studded With four boundaries and a six.
During their fifth-wicket partnership, Handscomb and Stoinis added 154 runs at just under three runs an over. It was the same pair that did the rebuilding job after Ojha three-wicket burst yesterday.
However, there was no contribution from the other batsmen as the visitors' innings folded up in 99.3 overs on the third day afternoon.
Karun Nair (4) and Shreyas Iyer (4) were at the crease at stumps.
India A had scored 301 all out in their first innings.
