The Virat Kohli-led side headed into Day Five in an advantageous position, needing only eight wickets with 129 runs to defend to clinch an innings win in the Test.
However, Marsh (53) and Handscomb (72*) displayed a fighting performance as the duo went on to share a gritting 124-run stand for the fifth wicket to help their side reach 204-6 in their second innings.
Earlier, India took a massive first innings lead of 152 runs as they ended their first innings at 603-9, thanks to 199-run stand between Cheteshwar Pujara (202) and Wriddhiman Saha (117).
While Saha smashed his third century, Pujara went on to score his third double hundred.
Pujara was declared 'Man of the Match' for his brilliant double ton.
With the draw, the four-match Test series remains levelled at 1-1.
The two sides will now lock horns in the final match in Dharamsala beginning March 25.
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