The hosts were cautious in reaching 54 for one before lunch in Hyderabad, replying to Australia's 237 for nine declared, but it was a much more fluid batting display in the afternoon as they put on 106 without loss in the session.
Following Michael Clarke's surprise declaration on Saturday, India began the morning with five runs on the board and played defensively. After underperforming with the bat, Australia knew they needed to make early inroads and their hopes were raised when Virender Sehwag fended behind off Peter Siddle in the fifth over of the day to depart for six.
India did not seem unduly worried, though, as Vijay drove James Pattinson for four in the next over and Pujara clipped his first ball to the boundary.
Thereafter India dug in before the break and it was 18 overs until Vijay struck the next boundary, a lofted drive off Xavier Doherty.
Although Doherty did beat the bat on occasion, the closest Australia came to another breakthrough was a run-out appeal in which replays showed Pujara was comfortably home.
They stepped it up a gear after lunch, with Pujara's intentions particularly clear as he struck three boundaries off a single Pattinson over early in the session while Vijay soon hit Doherty for six.
Test debutant Glenn Maxwell's first over was a maiden and helped Australia stem the tide, but he gave up 31 off his next six overs as India dominated.