Left-arm seamer Johnson removed Virat Kohli, Ajinkya Rahane and Rohit Sharma to record figures of 3-10 from 11 balls during a blistering spell of seam bowling, before latterly ejecting Umesh Yadav as India were dismissed for 224.
Australia made hard work of chasing their target of 128, with Ishant Sharma claiming three wickets and Yadav two, but Chris Rogers (55) held the innings together and Mitchell Marsh hit the winning runs to put the hosts 2-0 ahead in the four-match series.
India began day four on 71-1 but were dealt a blow before play with opener Shikhar Dhawan unable to take to the field first up having injured a wrist injury while batting in the nets.
Johnson quickly pounced on new batsman Virat Kohli, who had joined Cheteshwar Pujara at the crease, bowling the 25-year-old (1) with India having added just five runs to their overnight tally.
The Australia quick then had Rahane (10) and Rohit Sharma (0) pouched by Nathan Lyon and Brad Haddin respectively and when Josh Hazlewood trapped MS Dhoni lbw for a third-ball duck, India were teetering on 87-5, still 10 runs shy of making the Baggy Greens bat again.
Pujara (43) and Ravi Ashwin (19) lifted India into treble figures, while Dhawan reached 81 after returning to the crease at the fall of the seventh wicket, and tail-ender Yadav pummelled a quick-fire 32 off 42 deliveries as the tourists at least passed the 200 mark.
Australia looked nailed on for victory when they came out to bat with over a session left on day four and all of day five should they need it, but were made to work hard for their triumph.
Ishant Sharma sent David Warner (6) and Shane Watson (0) packing before tea, both batsmen edging behind to wicketkeeper Dhoni, but Rogers??? brisk 55 off 57 balls, which included 10 boundaries, seemingly eliminated any real danger of Australia capitulating.
The veteran opener became Sharma???s third victim with the score on 85 and Shaun Marsh (17) captain Steven Smith (28) and Haddin (1) were also banished to the hutch courtesy of Yadav, but Mitchell Marsh (6no) and Johnson (2no) saw Australia home.