Three wickets apiece from Hardik Pandya and Kuldeep Vadav left India in need of 270 to win against Australia during Wednesday's series-deciding third and final one day international in Chennai.
Australia were bowled out for 269 in the 49th over after Pandya (3-44) and left-arm spinner Yadav (3-56) put a brake on the tourists' early momentum.
But Pandya took three quick wickets -- Head on 33, Smith for a duck and Marsh on 47 -- to leave Australia worried at 85-3 by the 15th over.
Yadav toppled the middle-order beginning with David Warner, who made 23 on return from the elbow fracture that forced him out of the second Test in February.
Marnus Labuschagne (28) was next but Alex Carey stubbornly held on for another 10 overs before his departure on 38.
Fellow left-arm spinner Axar Patel and frontline pacer Mohammed Siraj each took two wickets apiece as Sean Abbott (26), Ashton Agar (17) and Mitchell Starc (10) stretched out the remainder of the innings.