The tourists were bowled out for 408 on Thursday, with Josh Hazlewood taking 5-68, before reducing the hosts to 221-4.
Although Australia are only 187 runs behind, Ashwin - who made 35 - has backed India to take control of the match if they perform well on Friday.
"We are ahead in the game at this point of time," he said. "If we can have a good morning session tomorrow then we will be ahead in the game overall.
"I think we could have scored a few more runs. I am disappointed how I got out. I think I batted to the best of my ability. After a long time I got a chance to bat in the middle, but felt disappointed to give my wicket away.
"We still got 400 runs but if we got a few more runs then it would have been better."
Australia captain Steve Smith will start day three unbeaten on 65, but Ashwin insists he is not their only focus as they look to take a lead into the second innings.
He added: "We are not concentrating on any one particular name. We are targeting all batsmen, top or lower order.
"On this sort of pitch it is easier to use your feet against the spinners, which Steve Smith did well.
"If it was a slightly difficult wicket with more footmarks, maybe I would have fancied myself more against him.
"Having said that I still fancy my chances against him [Smith] when play resumes."