Handscomb (72no) and Marsh (53) joined forces at a highly-vulnerable 63-4 when their captain Steve Smith fell shortly before lunch in the third Test.
But still 89 runs away from making their hosts bat again at that stage, they responded with a stand of 124 which shut India out for more than 62 overs on the way to 204 for six at stumps as the stalemate was agreed 10 minutes from the scheduled close.
The tourists had resumed initially on 23 for two on the last morning and, after an encouraging start, in the space of four balls lost Matt Renshaw lbw to Ishant Sharma from round the wicket and then Smith - to a misjudgment, bowled off-stump trying to pad away Ravi Jadeja.
The left-arm spinner finished with figures of four for 54 from a marathon 44 overs, and nine wickets in the match.
But Australia's fifth-wicket pair held sway. They each had a DRS lbw scare shortly before their 50s, Handscomb surviving on 44 against Umesh Yadav and Marsh on 48 against Ravi Ashwin.
Otherwise they were chanceless, Handscomb finishing unbeaten after keeping out 200 balls in all.
Despite Marsh's eventual departure to a short-leg catch off Jadeja and Glenn Maxwell's bat-pad to Ashwin, both teams can therefore still win the series when it concludes in Dharamsala next week.