Rohit Sharma, labouring his way to 56 off 89 balls, saw four of his teammates succumb to Australian bowlers' discipline, before he himself lost focus dancing down the track to Zampa, only to lose his wicket and his bat in the process.
A 37-run win in the fifth and deciding ODI in Delhi was always in reach of Australia with a competitive score and tight bowling setting up the result. It was India's first loss in an ODI series at home since 2015 when they lost 3-2 to South Africa.
Major General Asif Ghafoor, the director-general of Inter-Services Public Relations, won the internet when he mercilessly trolled the Indian cricket team following their 3-2 thrashing at the hands of Australia.
Shikhar Dhawan fell in the fifth over of India's reply but once Rohit was stumped after charging out of his crease to Zampa, India lost their way.
But when Khawaja and Maxwell fell to nearly identical dismissals - caught at cover by Virat Kohli - and Handscomb followed close behind, the momentum stalled with Australia dropping to 7/229 in the 46th over. He added 53 runs for the second wicket with Rohit.
The extravagance cost both Pant and Vijay Shankar (16) their wickets as they went after Lyon and Zampa respectively.
A brisk 91-run stand between Jadhav (44 off 57) and Bhuvneshwar Kumar (46 off 54) inspired India to a late challenge but it was too little too late as Australia took the final wicket with the final delivery to claim a famous victory. Tough place to come and play cricket against a very good side; they beat us in Australia.
Earlier, opener Khawaja, who had not scored a century before arriving in India for this series, hit an exact 100 off 106 balls as he shared two big partnerships at the top of the order to set platform for the visiting side.
Australia claims historic one-day international series win in India
After winning the toss, Australia were limited to 272/9 - after Khwaja, Finch and Peter Handscomb had helped the team to a formidable 175/1 in 32.5 overs - losing their last eight wickets for just 97 runs.
He hit 10 fours and two sixes on his way to three figures but his dismissal sparked a collapse for Australia.
Khawaja was yet again impressive with his footwork, handling the Indian spinners nicely before being caught at shot extra cover by Kohli off Bhuvneshwar.
Jadeja returned to get the unsafe Glenn Maxwell for one and Handscomb was caught behind off Shami as Australia slipped to 182-4.
Ravindra Jadeja replaced Yuzvendra Chahal in the Indian playing eleven while K L Rahul made way for Mohammad Shami.
The three confidence-building victories came after Australia had won just four of their past 26 completed one-dayers.
It turned out to be a forgettable day for chinaman Kuldeep Yadav, who bled 74 runs in his 10 overs, although he dismissed risky Ashton Turner (20) cheaply.