Wrapping up the win appeared a formality for India when their openers walked out at the picturesque venue nestled in the shadows of the Himalayas needing just 87 runs to seal the series 2-1 with all 10 second innings wickets intact.
While the miracle that Australia desperately needed did not materialise, paceman Pat Cummins injected brief excitement into the contest in his second over of the morning.
Murali Vijay, survivor of back-to-back lbw appeals from Josh Hazlewood, had looked a walking wicket all morning and the opener edged the first ball of that Cummins over to be caught behind for eight.
Five balls later, Cheteshwar Pujara belied his reputation as a good judge of singles and was run out without scoring when Glenn Maxwell, from point, threw down the middle stump at the non-striker's end.
Australia did not get another chance to celebrate as Rahul and Rahane combined to complete India's comprehensive victory with more than five sessions to spare.
Rahul made 51 not out, completing his sixth fifty in seven innings when he scored the winning runs at the Himachal Pradesh Cricket Association Stadium marking its debut as a test arena.
Leading the side in the absence of injured skipper Virat Kohli, Rahane smashed a rapid unbeaten 38 off 27 balls, an innings of uncharacteristic aggression that included back-to-back sixes off Cummins.
India all-rounder Ravindra Jadeja, who claimed 25 wickets to go with his couple of fifties, won the man-of-the-match and man-of-the-series awards.
The victory sealed India's seventh series win in a row, dating back to their triumph in Sri Lanka in August 2015.
The victory also brought a successful end to India's 13-test home season, in which they won 10, drew two and had the lone loss to Australia in the opener in Pune.
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