Australia, who began their second innings 152 runs adrift of the hosts, were in deep trouble after being reduced to 63-4 in Ranchi before Handscomb (72 not out) and Marsh (53) put on 124 for the fifth wicket.
The tourists eventually reached 204 for 6 at stumps on the fifth and final day, after the hosts made a mammoth 603-9 declared in their first innings.
India's Ravindra Jadeja claimed four wickets to return with overall match figures of 9-178 at India's newest Test venue.
But it was Cheteshwar Pujara (202) and Wriddhiman Saha (117) who were the main stars of the match for India after their 199-run seventh-wicket stand on Sunday.
The left-right batting combine of Handscomb and Marsh outmanoeuvered the Indian bowlers during their 232-minute resistance which was broken late in the final session.
Jadeja dismissed Marsh and his fellow spinner Ravichandran Ashwin then claimed the wicket of first innings centurion Glenn Maxwell for two, giving India brief hope of a dramatic final twist in the tale.
But a composed Handscomb played out the remaining few overs with Wade for company as Australia avoided defeat in their 800th Test.
Jadeja, who claimed the prized scalp of skipper Steve Smith (21), caused problems for all the Australian batsmen after the tourists resumed the day on 23-2.
Fast bowler Ishant Sharma trapped Matt Renshaw lbw for 15 in the morning session after an altercation between the two players seemed to have affected the left-handed batsman's concentration.
But it was Smith's wicket that brought a raucous home crowd to its feet after a quiet first hour of play.
Smith, who scored an unbeaten 178 in the first innings, tried to pad away a vicious turning delivery from Jadeja but it spun from outside leg to rattle the right-handed batsman's off stump.
Australia's comeback paceman Pat Cummins claimed four wickets, including that of Virat Kohli (6) who has managed just 46 runs in the series.
Kohli, who injured his shoulder while fielding on the opening day, did not take the field for the rest of Australia's innings but batted at his usual number four position.
Constant run-ins between the players of the two top sides continued to enliven the intense rivalry that reached boiling point in the previous game when Kohli accused Smith of abusing the decision review system.
The two teams now head to Dharamsala for the fourth and final match starting March 25.