Australia, who began their second innings 152 runs adrift of India, were in deep trouble after being reduced to 63-4 before Peter Handscomb (72 not out) and Shaun 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 in Ranchi.
Ravindra Jadeja claimed four wickets to return overall match figures of 9-178 at India's newest Test venue. But Smith said Australia would go into next weekend's series finale in Dharamsala full of confidence.
"If there's anything called momentum in cricket, it's with our side at this moment," Smith told reporters after the match.
"India coming today would have expected to bowl us out. I'm sure they're hurting a little bit in their change room. Having said that, it's (four-match series) 1-1 and looking forward to the decider."
Cheteshwar Pujara (202) and Wriddhiman Saha (117) were the main stars of the match for India with a 199-run seventh-wicket stand on Sunday, while Smith himself scored an imperious unbeaten 178 in Australia's first innings.
But it was Handscomb and Marsh's 232-minute resistance on the final day -- broken only late in the last session -- that proved decisive.
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.
"I'm proud with the way the boys were able to stick it out today," said Smith.
"The way Marsh and Handscomb batted today was fantastic. I thought they played beautifully. They didn't look like they were getting out for a while. I'm really proud of them.
Jadeja, who claimed the prized scalp of Smith for 21 second time round, 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.
He tried to pad away a vicious turning delivery from Jadeja but it spun from outside leg to rattle the right-hander's off stump.
Speaking at the post-match press-confrence, Kohli praised Jadeja and Pujara for putting India in a strong position after Australia's first-innings 451.
"Jadeja's bowling was outstanding. He was high-class and proved why he is the joint top Test bowler along with Ashwin," said Kohli.
On man of the match Pujara, who registered his third Test double century, Kohli said: "He is the most composed player in the team... someone like that is priceless to have him in the team."
Australia's comeback paceman Pat Cummins claimed four wickets, including that of captain 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 kept alive the intense rivalry that reached boiling point in the previous game, when Kohli accused Smith of abusing the decision review system.
Australia won the first Test in Pune before India came back to level the series in the ill-tempered match in Bangalore.