The tourists will start the final day's play on Monday with an overall lead of 263 with five wickets in hand on a deteriorating wicket that is making batting difficult.
Ponting, whose team closed the fourth day at 193-5 in their second knock, has three sessions left in the match to force a result by giving his bowlers enough time to dismiss India again.
India not only prolonged their first innings to 360 to narrow Australia's lead to 70 runs, but slowed their rivals' bid for quick runs by grabbing wickets at regular intervals.
Zaheer Khan led India's fightback with an unbeaten 57, his team's top score, and then removed the dangerous Matthew Hayden for 13 when the Australians batted a second time.
Hayden was given out leg-before to Zaheer by Pakistani umpire Asad Rauf for the second time in the match, but TV replays indicated the ball would have missed the leg-stump.
Ponting, who made a century in the first innings, was snapped up low at short mid-wicket by Venkatsai Laxman off seamer Ishant Sharma for 17.
Opener Simon Katich (34) put on 50 for the third wicket with Michael Hussey when he was caught by Laxman at silly point off Harbhajan Singh to make the tourists 99-3.
Australia were just 198 ahead when the fifth wicket fell, Hussey bowled by a Harbhajan 'doosra' -- the ball that turned into the left-hander instead of going away from him.
Shane Watson (32) and Brad Haddin (28), however, kept the tourists on target by adding 65 for the unbroken sixth wicket.
India were hampered in the field by a shoulder injury to captain and spin spearhead Anil Kumble, who was unable to bowl till the 54th over of the innings.
India, who started the day at 313-8, batted till 30 minutes before lunch, before last man Sharma was bowled by part-time spinner Michael Clarke for six.
Zaheer had on Saturday put on 80 runs for the eighth wicket with fellow tailender Harbhajan, who launched the home team's late charge with a stroke-filled 54.
India's last five wickets added a valuable 205 runs after the top five had been dismissed with just 155 on the board.
Zaheer batted for three hours to compile his second Test fifty with the help of seven boundaries.
India's total was helped along by a whopping 52 extras conceded by the Australians, including 23 byes by wicket-keeper Brad Haddin on the unpredictable pitch.
They were the highest number of extras given away by an Australian team against India in an innings, surpassing the 45 Allan Border's tourists conceded in Mumbai during the 1986 series.
Kumble, who came in on Saturday evening at the fall of Harbhajan's wicket, put on 31 for the ninth wicket with Zaheer before he was leg-before to Watson for five.
Watson finished with 3-45, while left-arm fast bowler Mitchell Johnson failed to add to Saturday's tally of four wickets.