Australia dumped frontline slow bowler Nathan Lyon after the hammering the offspinner received in India's eight-wicket win in the first Test in Chennai, where Lyon took 3-215 and 1-29.
Left-arm quick Mitchell Starc was also axed.
Their replacements, debutant offspinner Glenn Maxwell (0-55 off 10 overs) and three-Test spinner Xavier Doherty (0-85 off 26), were unable to break through.
In Doherty's defence he did have Vijay dropped at slip on 71 and Pujara edged through slip on 157.
Pujara and Vijay's 294-run stand is a record for India for the second wicket against Australia.
Former India allrounder Ravi Shastri questioned Australia's selection strategy.
"What kind of message does it send to Nathan Lyon to be dropped after taking four wickets in Chennai?'' the 80-Test veteran asked.
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"I can't imagine this would be good for Nathan Lyon's confidence.
"He came here as the frontline spinner and then he is upstaged in this Test by Doherty. It is a big call and unfortunately it could backfire for Australia,'' the TV commentator added.
Skipper Michael Clarke, a part-time spinner who has a long-term back problem and arrived in India earlier this month nursing a sore hamstring, didn't bowl.
Siddle struck in the fifth over of the day when Virender Sehwag was caught behind for six and Australia limited India to 1-54 at lunch before Pujara and Vijay cut loose.
India lead the four-match series one-nil.
- AAP