The tourists, replying to the Indian Board President's XI total of 455, were bowled out for 314 in their first innings before tea on the third day at the Rajiv Gandhi stadium.
The hosts were 110-2 in their second knock at stumps and will begin the final day on Sunday with an overall lead of 251 runs with eight wickets in hand.
Test discards Wasim Jaffar (46) and Yuvraj Singh (37) were at the crease, having put on 75 for the unbroken third wicket.
The match is Australia's final warm-up before the four-Test series opens in Bangalore next Thursday.
Australia crashed from their overnight score of 191-4 to 218-9 before last man Stuart Clark proved an unlikely saviour by adding a valuable 96 runs for the 10th wicket with the reliable Mike Hussey.
Left-handed Hussey showed his team-mates the way to tackle spin on Indian wickets with a gritty 126 that contained 14 boundaries and a six.
Clark, who averages 11.80 with the bat in 18 Tests, struck four boundaries and two sixes in a defiant 44 before holing out to mid-wicket against home team captain Yuvraj.
The 19-year-old Chawla and spin partner Pragyan Ojha shared five wickets in the space of 19 runs to slice through Australia's lower order in dramatic fashion in the morning session.
The tourists had added eight runs to their overnight score when left-armer Ojha had Brad Haddin stumped for 34, ending a 71-run stand for the fifth wicket with Hussey.
Ojha dismissed Jason Krejza before Chawla, ignored by the selectors for the first two Tests, gave a reminder of his wicket-taking potential.
The leg-spinner added to his two wickets on Friday by removing Brett Lee, Mitchell Johnson and Peter Siddle in the space of five overs.
The hosts slipped to 35-2 by the 13th over in their second innings as Clark trapped Aakash Chopra leg-before and Lee had Subramaniam Badrinath caught behind by Haddin.
Jaffar struck five boundaries in his unbeaten knock, while Yuvraj slammed two sixes off Michael Clarke just before close.
Board President's XI - 1st innings 455 (R. Sharma 105, V. Kohli 105, P. Patel 62, Irfan Pathan 56, M. Johnson 4-75, S. Clark 3-67)
Australia - 1st innings (overnight 191-4):
S. Katich c Kohli b Ojha 15
M. Hayden c Patel b Pathan 20
R. Ponting b Chawla 41
M. Hussey not out 126
M. Clarke lbw b Chawla 18
B. Haddin st Patel b Ojha 34
J. Krejza c Yuvraj b Ojha 0
B. Lee lbw b Chawla 0
M. Johnson c Chopra b Chawla 5
P. Siddle lbw b Chawla 2
S. Clark c Ojha b Yuvraj 44
EXTRAS: (b4, lb5) 9
Fall of wickets: 1-26 (Hayden), 2-54 (Katich), 3-94 (Ponting), 4-128 (Clarke), 5-199 (Haddin), 6-203 (Krejza), 7-204 (Lee), 8-214 (Johnson), 9-218 (Siddle), 10-314 (Clark).
Board President's XI - 2nd innings:
A. Chopra lbw b Clark 6
Wasim Jaffer not out 46
S. Badrinath c Haddin b Lee 14
Y. Singh not out 37
EXTRAS: (b4, nb3) 7
TOTAL (for two wickets) 110
Fall of wickets: 1-7 (Chopra), 2-35 (Badrinath)
Lee 9-1-30-1 (2nb)
Clark 5-2-10-1 (1nb)
Board President's XI leads by 251 runs with eight wickets in hand.