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.
Nearly all of Australia's frontline batters got the starts in the match but only David Warner (52) managed a half-century.
Australia still looked poised for a bigger total with Josh Inglis (45) and Marcus Stoinis (29) scoring freely.
After Adam Zampa removed both the openers and Shreyas Iyer ran himself out in between, Australia looked back in the contest.
Rahul made 58 not out, staying put till the end to secure India's victory by hitting Abbott for a four and a six.
Persons:
India's Mohammed Shami, Andrew Couldridge, Seamer Mohammed Shami, Mitchell Marsh, KL Rahul, David Warner, Steve Smith, Marnus Labuschagne, Cameron Green, Josh Inglis, Marcus Stoinis, Shami, Shubman Gill's, Ruturaj Gaikwad, Jos Inglis, Sean Abbott's, Gaikwad, Rohit Sharma, Adam Zampa, Shreyas Iyer, Suryakumar Yadav, Rahul, Abbott, Amlan Chakraborty, Christian Radnedge
Organizations:
Kia, Australia, KL, Thomson
Locations:
England, India, London, Britain, Australia, Indore, Rajkot, New Delhi