The visitors lost 7-69 in Ranchi as they limped to 8-118 before weather ended their innings with eight balls remaining.
After a delay of almost two hours, India were given six overs to chase 48 which they hunted down with three balls to spare.
Rohit Sharma made 11 off seven balls before Nathan Coulter-Nile found a way through his defences to claim the only wicket of the innings.
Indian skipper Virat Kohli (22 off 14) and opener Shikhar Dhawan (15 off 12) did the rest, cruising to 1-49 after smashing three boundaries each.
Australia will have to win Tuesday's match in Guwahati to avoid another series loss, after a deflating 4-1 drubbing in the one-day internationals.
Aaron Finch picked up where he left off in the ODIs, top-scoring with 42 from 30 balls.
"I thought after we got off to a pretty good start with the bat, obviously losing our way through the middle order again," Finch said.
"But it was a very challenging wicket to start on in particular against the spin.
"The guys were formulating a plan quite nicely, we just seemed to keep losing wickets at the wrong time."
He smashed five boundaries including a six before a crafty piece of bowling from left-arm wrist spinner Kuldeep Yadav turned him inside out and clattered into the stumps.
Yadav (2-16 off four overs) had his second victim when Moises Henriques (8) advanced down the wicket to a leg break which kept low, clean bowling him.
Stand-in skipper David Warner made eight, while the injured Steve Smith ran the drinks and will return to Australia on Sunday.
Glenn Maxwell couldn't play himself into form, struggling for his 17.
Maxwell pulled a long hop straight to mid-wicket off Yuzvendra Chahal who has claimed his wicket in all four of the Victorian's innings on the tour.
Australia's middle order again failed when they were needed, mimicking their efforts in the ODIs.
Travis Head (nine), Tim Paine (17) and Nathan Coulter-Nile (one) completed the sextet of Australians who were bowled.
"Especially after winning the toss and bowling first, that effort was required," Kohli said.
"We don't really understand the Duckworth-Lewis method. After getting them down to 118, we thought it would only be 40 or something. 48 was tricky."