MORE: Big Bash: Lunatics, louts and larrikins | AS IT HAPPENED: Second Test - Australia v India
Having been set 128 to win with more than four sessions to play on Saturday, Australia stumbled just before tea with David Warner (six) and Shane Watson (zero) dismissed by Ishant Sharma to give India slim hope of causing an upset.
But resuming at 25-2, Rogers ensured Australia would take a 2-0 series lead smashing consecutive boundaries in Umesh Yadav's first over after tea.
The veteran opener went on to post 55 before becoming Sharma's third victim of the innings but by then he had put Australia within touching distance of victory at 85-3, with Mitch Marsh (six) and Johnson (two) taking the hosts across the finish line after a nervous finish.
The penultimate day at the Gabba started in strange circumstances with Indian opener Shikhar Dhawan not taking his place in the middle after being struck on the wrist during his warm-up session.
As the Board of Control for Cricket in India complained about the standard of the Gabba's practice pitches, their team seemed distracted with Dhawan's replacement - Virat Kohli - chopping a delivery from Johnson onto his stumps in just the fourth over of the day to be dismissed for one.
Kohli's wicket was the first of five in 11 overs as India slumped to 117-6 - Johnson claiming three scalps - prompting Dhawan to resume his innings on 26.
Despite his earlier injury, Dhawan attacked Australia's bowlers - often monopolising the strike after Cheteshwar Pujara was dismissed for 43 - before eventually getting trapped lbw by Nathan Lyon for 81.
India added another 21 runs after Dhawan's innings ended to be all out for 224 with Johnson finishing with figures of 4-61 off 17.3 overs, while Lyon, Mitch Starc and Josh Hazlewood all claimed two wickets.
Australia had to survive a shaky finale after losing Rogers, Shaun Marsh (17), Steve Smith (28) and Brad Haddin (one) before Mitch Marsh struck the winning runs to claim a four-wicket triumph in Smith's first match as skipper.