Put into bat, Australia were all out for 276 in exactly 50 overs even though 300 looked within their reach before Shami polished off their tail.
In reply, four of India's top six batters smashed half-centuries as the home side led, who have rested several frontline players, romped home with eight balls to spare.
Shami removed Mitchell Marsh in the first over to give India a strong start after home captain KL Rahul won the toss and elected to field.
Nearly all of Australia's frontline batters got the starts in the match but only David Warner (52) managed a half-century.
The left-hander combined in a 94-run stand with Steve Smith, who made 41 in his return to action from a wrist injury.
Marnus Labuschagne (39) and Cameron Green (31) chipped in with useful runs in the middle order but could not soldier on.
Australia still looked poised for a bigger total with Josh Inglis (45) and Marcus Stoinis (29) scoring freely.
Shami returned to claim three lower-order wickets to hasten their collapse.
India began their chase brightly, aided by Shubman Gill's (74) silken batting and a reprieve to Ruturaj Gaikwad (71) when Jos Inglis dropped the batter, then on 17, off Sean Abbott's bowling.
Gaikwad opened the innings in place of regular skipper Rohit Sharma, who is among several frontline players rested for the first two matches of the series to keep them fresh for the home World Cup beginning on Oct. 5.
After Adam Zampa removed both the openers and Shreyas Iyer ran himself out in between, Australia looked back in the contest.
Suryakumar Yadav, who made a cheeky 50, and Rahul batted sensibly to frustrate Australia.
Rahul made 58 not out, staying put till the end to secure India's victory by hitting Abbott for a four and a six.
The teams move to Indore for the second match on Sunday and the three-match series concludes in Rajkot on Wednesday.
