Nathan Lyon struck in the second over of the day to get Rahane nudge one to short leg.
Pant scored a quickfire century to keep India in the game after being dropped twice by Tim Paine off Lyon.
Pant went on to share a century stand with Pujara to keep Asurtalia at bay.
Pant fell for 97, ending a superb stand with Pujara that hassled the Australians. Pujara went on to get his second fifty of the game.
Hazlewood cleaned Pujara up as Australia bounced back.
Hanuma Vihari picked up a hamstring injury but yet kept Australia at bay with a match-altering partnership with Vihari.
R Ashwin and Vihari batted out the remainder of overs as Australia couldn't take their chances to draw the third Test with the series still reading 1-1.
Ashwin and Vihari batted 258 balls to help draw the Sydney Test.