Clarke hit four fours and a six in his unbeaten 57-run knock and also shared a 73-run stand with all-rounder Shane Watson (36) to leave the tourists 177 runs adrift of India's first innings score of 613-7 declared.
Clarke's effort came on the back of useful contributions by Matthew Hayden (83), Simon Katich (64), Ricky Ponting (87) and Michael Hussey (53).
The Indian bowlers toiled on a wearing Feroz Shah Kotla track and were rewarded with the wickets of Watson and Brad Haddin (17).
But the hosts were sloppy in the field, highlighted by Ishant Sharma dropping a simple chance at mid-off when he failed to latch on to a pull by Clarke off leg-spinner Amit Mishra.
Clarke, then on 21, grew in confidence, lofting Mishra for a six over the mid-wicket fence en route to his 10th Test fifty.
Part-time off-spinner Virender Sehwag (4-83) added one more wicket to his overnight three scalps when he knocked Watson's leg stump with a sharply turning ball.
Indian skipper Anil Kumble (1-71) returned to bowl in the 15th over a day after needing 11 stitches for a deep cut he sustained on his left little finger after lunch on Friday.
Kumble, who once bowled with a broken jaw in Antigua, picked his first wicket of the series when he had Haddin stumped by Mahendra Dhoni.
The veteran bowler went wicketless in the drawn opening Test at Bangalore and missed India's 320-run victory in Mohali with a shoulder injury.
The fourth and final Test will be played in Nagpur from Nov 6-10.