Stoinis, who is in Alice Springs playing for Victoria in the Sheffield Shield, will fly to India to join up with the touring side when the match ends.
Marsh has been struggling with a shoulder injury and will return to Australia after a disappointing campaign in which he scored 48 runs in four innings and bowled a total of five overs in the first two tests at Pune and Bengaluru.
Stoinis, who scored an unbeaten 146 against New Zealand in the Chappell-Hadlee ODI series last month, played seven games for the Kings XI Punjab in India's domestic Twenty20 league last year.
The 27-year-old, who also toured India with Australia A in 2015, is not guaranteed to replace Marsh in the starting XI for the next test with spin-bowling all-rounders Ashton Agar and Glenn Maxwell already at coach Darren Lehmann's disposal.
The series stands level at 1-1 after India's dramatic 75-run victory in Bengaluru on Tuesday with the penultimate test starting in Ranchi on March 16.
(Reporting by Simon Jennings in Bengaluru, editing by Nick Mulvenney)
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