Pujara raised his 21st test half-century to be 68 not out at stumps, with Kohli on 47 as Australia's bowlers endured a day of toil on a toothless Melbourne Cricket Ground wicket after the India captain won the toss and elected to bat.
Opener Mayank Agarwal enjoyed a stellar introduction to test cricket, scoring 76 to record the highest ever innings by an India debutant on Australian soil before being dismissed on the last ball before tea.
The Karnataka batsman eclipsed the 51 scored by Dattu Phadkar against Don Bradman's team in Sydney in 1947, raising his hand for a permanent role in the top order after replacing the out-of-form Murali Vijay.
The Boxing Day crowd of 73,516 was a record attendance for a match featuring India in Australia and travelling fans from the subcontinent roared their applause as Kohli and Pujara combined for a 92-run stand and strode off unbeaten at stumps.
A day after Christmas festivities, local supporters were left flat, however, and the Australian bowlers frustrated by the drop-in pitch that offered very little movement.
With the four-match series poised at 1-1, India's disciplined performance left them well-placed to build a big total and tire Australia's bowlers in baking heat forecast for day two.
Only Pat Cummins was rewarded for Australia, the paceman claiming the wicket of new opener Hanuma Vihari and ending Agarwal's 161-ball knock by having him glove a catch down the legside.
That was the sole highlight until the brink of tea for Tim Paine's side, with Agarwal and Pujara holding firm in an 83-run stand.
With the pitch doing little for the seamers, spinner Nathan Lyon was introduced in the eighth over of the morning but he was also unable to conjure a breakthrough.
Agarwal smashed two fours past the bowler after lunch, the second a glorious straight drive to raise his fifty off 95 balls.
He slogswept Lyon for six after the drinks break before eventually falling to Cummins at 123 for two.
Kohli joined Pujara in the middle and feasted on the ageing ball, punching the ball to the fence with a trio of exquisite, wristy on-drives among his six fours.
Desperate for a wicket, Australia threw away a review for lbw when Lyon rapped Kohli's pads when the captain was on 32.
The appeal was turned down and the review squandered, with the replay showing the ball pitching a long way wide of off-stump.
After the second new ball was taken, a more realistic chance went begging when Kohli, on 47, nicked a Mitchell Starc delivery only for wicketkeeper Paine to grass a low, one-handed chance.
