The hosts began the final day on 258-8 chasing an unlikely 399, but after rain delayed the start, India needed only 4.3 overs to wrap up the win.
Jasprit Bumrah, who took career-best match figures of 9-86, had Pat Cummins edge to slip for 63, before Nathan Lyon gloved Ishant Sharma behind.
The fourth and final Test starts in Sydney on 2 January.
Bumrah's haul was the best by an India pace bowler in Australia and the 25-year-old leads the series standings with 20 wickets, three more than Australia spinner Lyon.
"Credit has to go to the bowlers. I think they have been outstanding, especially Jasprit. The way he bowled in this Test match was outstanding," said India captain Virat Kohli, whose side will retain the Border-Gavaskar Trophy.
"The mindset he has at the moment is what separates him from anyone else in the world right now. He looks at a pitch and he doesn't think it's a hard toil, he thinks wickets, he thinks 'how can I get a breakthrough for the team'.
"What I've seen in the last 12 months is the way he has matured in Test cricket and the way that he is bowling so soon, so quickly in his Test career, I think is a very scary sign for all the batsmen around the world.
"I mean, in Test cricket if there is a pitch like Perth, I wouldn't want to face Jasprit Bumrah, to be honest, because once he's on a roll he can really crank it up."
India bounced back after losing the second Test in Perth to declare twice at the MCG, with their bowlers tearing Tim Paine's side apart for 151 and 261.
The defeat in Melbourne came in Paine's sixth Test match as Australia captain after replacing Steve Smith, who was banned along with David Warner and Cameron Bancroft for ball-tampering during the third Test against South Africa in Cape Town in March.
Smith and Warner's year-long suspensions will expire in late March 2019, and Paine admitted Australia missed their presence following a poor batting performance.
"I think it's inexperience, it's pressure, India's bowling attack is probably as good as most of us have ever played," he said.
"I think it's pretty clear that if you take two or three of the best players out of anyone's line-up around the world, you're going to struggle a little bit at times and you are going to have inconsistent performances.
"The silver lining is that we do have world-class players that are available soon to come back into this side, and clearly when they do they will make a huge difference."
With several players under pressure for their place, the Aussies have added all-rounder Marnus Labuschagne to their squad for the final Test.
Meanwhile, Bancroft made only two, a day after his nine-month ban expired, as Perth Scorchers lost to Hobart Hurricanes in the Big Bash.
Australia v India: Tourists win by 137 runs
Third Test, Melbourne (day five)
India 443-7 dec (Pujara 106) & 106-8 dec (Cummins 6-27)
Australia 151 (Bumrah 6-33) & 261 (Cummins 63, Bumrah 3-53)
India won by 137 runs; lead series 2-1