Pujara was unbeaten on 162 while Vijay was not far behind on 129 after a day of delight for the watching India fans in Hyderabad.
They had begun slowly after Michael Clarke's surprising declaration on 237 for nine on Saturday, resuming on five without loss and playing 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 made it through to lunch on 54 for one, but then stepped it up a gear with Pujara's intentions particularly clear.
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 the rest of the session.
They added 106 runs to reach tea on 160 for one, but were still only warming up and the final session of the day saw them pile on 151 more, their unbroken partnership worth 294 by the close.
Australia looked worryingly short of wicket-takers, with Doherty the only man to even slightly unsettle either batsman as he found a little turn, but nothing they were not able to handle.
The tourists will need to find a way to fix that on Monday if they are not to let this Test get away from them as the series opener did in Chennai.