Handscomb (72 not out) and Marsh (53) forged a brilliant 124-run fifth-wicket stand as Australia ended at 204/6 and denied the hosts any chances of gaining a 2-1 lead in the series.
Both the middle order batsmen showed patience and skills to stitch a partnership of 124 runs for the fifth wicket. The visitors scored 451 & 204/6 in 100 overs in response to India's 603/9 (declared).
Ravindra Jadeja was the pick of the Indian bowlers who grabbed 4/54 as he did the bulk of the bowling for his side. Ishant Sharma (1/30) and Ravichandran Ashwin (1/71) got one wicket apiece.
Credit goes to Australia for saving the match who started day's proceedings from 23/2. The visitors went on to add another 181 runs and lost just 4 wickets.
With this series levelled 1-1, the two teams would lock horns in the 4th and final Test which starts in Dharamsala on March 25.
Cheteshwar Pujara was adjudged the man of the match for his mammoth 202 run knock. He played 525 balls in his innings to notch up his third Test double.Handscomb consumed 200 deliveries and remained unbeaten on 72.
Marsh produced his own impersonation of a stone wall at the other end, taking up 197 deliveries enroute to an extremely patient 53 before falling to an excellent catch by Murali Vijay off Ravindra Jadeja's bowling around nine overs from the close.
Off-spinner Ravichandran Ashwin got rid of Glenn Maxwell a couple of overs later, but that proved to be the last bit of meaningful action in the match as Matthew Wade joined Handscomb to bat out the remaining overs.
For India, left-arm spinner Jadeja was the pick of the bowlers with figures of 4/54 while Ashwin and pacer Ishant Sharma clinched a wicket each.
Cheteshwar Pujara was adjudged the man of the match for his double century during India's first innings.
Pujara had scored 202 runs off a marathon 525 deliveries on the fourth day to create the record of most balls faced in an innings by an Indian batsman.
He is the first Indian to face more than 400 balls in a Test innings at home since V.V.S. Laxman batted 452 deliveries during his 281-run knock against Australia during the Kolkata Test in 2001.
The four Test series remains level at 1-1.
Resuming on the fifth and final day at the precarious position of 23/2, Matt Renshaw and Australian captain Steven Smith batted steadily in an effort to carry the visitors to safety.
Both batsmen simply tried to bat as long as possible, taking occasional scoring opportunities but not taking any risks whatsoever.
They managed to bat out the first hour of the day before Ishant Sharma and Jadeja took two wickets in a space of four balls to keep the pressure firmly on Australia.
India captain Virat Kohli began with Jadeja at one end and fast bowler Umesh Yadav at the other. Smith left plenty of balls alone as the plan seemed to be getting his edge outside off-stump.
Renshaw used his height and big stride on the front foot defence in his efforts to nullify Jadeja's spin even as the left-arm spinner tried to utilise the rough just outside the left-hander's off-stump.
Ishant soon replaced fellow pacer Yadav and found a hint of movement from around the wicket.
Renshaw's decision to take on Sharma from the first ball of the 29th over of the innings drew the bowler's ire, and he hurled down a pair of bouncers in the same over to push the opener back, before trapping him plumb in front for the lbw decision.
In the very next over, Smith failed to get his front leg far enough down the pitch and and also erred in picking out the line of a Jadeja delivery only to see his off-stump get knocked over.
Smith used up 68 deliveries while scoring his 21 runs while Renshaw scored 15 runs off 84 balls.
The patient, gritty display by both batsmen set the tone for the rest of Australia's second innings as Marsh and Handscomb knuckled down in a bid to save the Test.
(With inputs from agencies)