India faced three overs late on Day 1 after Australian captain Michael Clarke declared at what appeared to be a below par 9(dec)-237.
Both Virender Sehwag and Vijay survived to enter Day 2 with the score 0-5, and after Sehwag fell for six early on Sunday, Vijay and Pujara had to fight hard to survive until lunch.
The going was slow with the pair scoring just 37 runs from 132 balls, but the survival was the key and now the pair went on to bat right throughout the rest of the day's play to make Australia's bowling attack look decidedly second-rate.
Both players brought up their centuries during the final session of the day's play and by stumps India had reached 1-311 to be 74 runs ahead of Australia with nine wickets remaining and three days of the Test to go.
Vijay has now moved along to 129 and Pujara 162 as well with the pair so far combining for an unbeaten partnership of 194. With Sachin Tendulkar, Virat Kohli and MS Dhoni still to come for India, Australia could be facing a massive task to even avoid an innings defeat from here.
Australian pacemen Peter Siddle and James Pattinson found life much more encouraging on the Hyderabad pitch than in Chennai, and kept things nice and tight right throughout the first session on the second day.
That led to the wicket of Sehwag for six when he was caught behind for six by Matthew Wade off the bowling of Siddle.
Neither Vijay nor Pujara found it easy to score runs leading into lunch, but they survived and the going got a lot easier after the luncheon interval with Australia's attack hardly creating a chance.
Pattinson (0-56 from 18 overs), Siddle (1-51 from 19) and Moises Henriques (0-38 from 19) worked hard and kept things reasonably tight, but the pitch has now dried out even more and offering them less and less as the match goes on.
Left-arm spinner Xavier Doherty has looked solid with 0-85 from 26 overs and was unlucky not to pick up the wicket of Vijay when he was dropped at slip by Clarke on 73.
Australia's selectors clearly got things wrong in Chennai, but it now appears they have again by opting to pick Glenn Maxwell in favour of Nathan Lyon.
At this stage of his career, debutant Maxwell appears nothing more than a part-time option and already has been easy for Vijay and Pujara to handle, going for 0-55 from 10 overs.
The combination of the left-arm spin of Doherty and off-spin of Lyon likely would have been Australia's best option of being competitive in Hyderabad.