At the break, Vijay was batting on unbeaten 71 that came off 156 balls with the help of five boundaries and two sixes, while giving him company was skipper Virat Kohli on 26 off 56 balls during which he hit two fours.
The duo had put on 56 runs for the third wicket with India still needing another 189 runs in the last session for victory after Australia had already pocketed the four-Test series 2-0.
For Australia, Nathan Lyon (1/81) and Shane Watson (1/15) were the wicket-takers while Josh Hazlewood (0/6) bowled a brilliant spell of nine overs without any success.
Coming on to bat after lunch, Vijay and Rohit Sharma (39 runs, 90 balls, 2 fours, 2 sixes) looked to carry on their good work, but their partnership didn't last long.
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Rohit came out positive but perished when he chose to guide one from Watson down past the slip cordon and Australia captain Steve Smith took a brilliant one-handed catch diving to his right to dismiss the batsman.
The 100-run mark for India came up in the 37th over and then Vijay had a huge moment, when on 42, he was dropped by Shaun Marsh at short cover off Ryan Harris.
In the very next over, on 46, Vijay survived another LBW shout as umpire Richard Kettleborough was not convinced even though TV replays showed that he was out.