At the tea break, JP Duminy (15 batting) and wicketkeeper Dane Vilas (2 batting) were at the crease, still needing 159 runs for a series-levelling win.
Mishra struck twin blows in a space of six deliveries in successive overs in the post lunch session to put a spanner on the slim South African hopes after Amla (39) and Du Plessis (39) stonewalled the Indian bowling with a fifth wicket stand of 72 runs.
Amla, who had scored 253 at this ground in a winning cause five years ago, had kept the Indians at bay for 219 minutes and hit just 2 fours in his 167-ball knock.
Du Plessis, who showed admirable patience till his skipper's departure, lost it five balls later when he played an ill-advised pull shot to a ball that kept low from Mishra and hit the middle stump.
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The home team's bowlers were kept at bay by the stone-walling tactics of Amla and Du Plessis resulting in the post-lunch hour of play producing a measly 21 runs in 19 overs.
The duo, in the process, came up with the longest partnership of the series overtaking the one between Murali Vijay and Cheteshwar Pujara in the series opener at Mohali.
Earlier, Ravindra Jadeja's stroke-filled half-century enabled India to get the all-important lead.
Jadeja (63 off 95 balls) reached his seventh half-century in Tests with the help of four boundaries and four sixes.
The seventh wicket partnership of 96 runs in 30 overs with Saha (31) could well prove to be the turning point on this Dharamsala track.
For Australia, Nathan Lyon (5/98) had a five-wicket haul. Saha, on his part again showed steely temperament despite another testing morning spell by Patrick Cummins (3/94) and Josh Hazlewood (1/51).
Jadeja was adjudged caught behind off the very first ball of the third morning bowled by Cummins but he promptly used DRS to get a favourable decision after the TV replays confirmed that there was no edge.
His best shot off the morning was a picture perfect straight drive off Cummins that raced through grass blades.
The moment, Lyon flighted one, he jumped out to loft him effortlessly over long-off for his third six. He also had a streaky boundary off Lyon past the slip cordon.
While Saha dabbed around for singles and occasional doubles, Jadeja took risks at will.
A quick single brought up his 50 and there was customary sword wielding celebration from the man. In fact, Jadeja joined Virat Kohli, KL Rahul and Murali Vijay among others, who have surpassed 50 six times in the current season. He also became the second Indian cricketer after Kapil Dev to achieve the double of 500 runs and 50 wickets in a season.
Cummins had the last laugh when he played on trying to smash a fullish delivery. Bhuvneshwar Kumar's (0) was Steve O'Keefe got his first victim of the match.
Cummins then got Saha with an unplayable delivery that climbed big on him as he tried to awkwardly fend it. The ball popped up off his gloves and Steve Smith backpedalled and timed his jump to perfection to take a spectacular catch.