Finch scored a run-a-ball 107 and posted an opening partnership of 187 runs with David Warner (93) to lay the foundation of a big total.
Steven Smith (51) also posted a useful contribution in the middle order before Glenn Maxwell (41) came up with the fireworks towards the end to give the Australian innings a superb finish.
Pacers Ishant Sharma (4/77) and Umesh Yadav (3/67) came up with a flurry of late wickets to keep the Australian total below the 350-run mark.
India had batted first in the three previous matches, losing by five wickets, seven wickets and three wickets respectively to give Australia a decisive 3-0 lead in the five-ODI series.
Electing to bat after winning the toss, Australia skipper Steven Smith saw his decision vindicated as openers Finch and Warner tormented the Indian bowling.
Yadav started off on a promising note, bowling a maiden in the opening over of the Australian innings and smiled to himself as he walked over to his fielding position. That was one of the few occassions the Indian bowlers could afford to smile over the course of the next 49 overs.
Bhuvneshwar started the next over with a full length delivery which drifted onto the legstump. Finch promptly glanced the ball off his pads to dispatch it to the boundary and kick-off the Australian run spree.
Warner, who replaced Shaun Marsh in the Australian XI after returning from paternity leave, was the more aggressive among the Australian openers. The left-hander smashed 12 boundaries and a six during his 92-ball knock.
Finch gave Warner good support from the other end as the hosts kept the scoreboard ticking at healthy rate, helped on by some inconsistent fielding performance by the visitors.
The Australian blitzkrieg claimed a victim in umpire Richard Kettleborough when the Englishman was hit on the right leg by a Finch straight drive in the 17th over. Kettleborough limped off the field and was replaced by Australian official Paul Wilson.
Warner was looking well set for a well-deserved century before being bowled while trying to hoist a slowish, incoming delivery from Ishant over mid-wicket.
But that proved to be no respite for the visitors as Finch soon brought up his century, playing Yadav to midwicket for a single. Finch finally sent back by Yadav when he mistimed an attempted pull shot. The ball found the upper edge of the bat and went up in the air as Ishant pulled off a comfortable catch.
Mitchell Marsh, promoted up the order to the number three slot, did not seem to be in his elements on the day. He produced a sxeries of ill-timed shots that mostly found the fielders before being dismissed by Umesh for a 42-ball 33.
But Indian hopes of containing the Australians to a manageable total soon disappeared as Smith and Maxwell smashed the bowlers all around the ground. Smith hit four boundaries and three sixes in a 29-ball 51 and Warner had six boundaries and a six in his 20-ball innings.
The Indian bowlers hit back late in the innings, claiming five wickets in the last six overs. But by then, the Australians had already ste themselves up for a formidable total.
The following is the scoreboard at the end of the Australian innings in the fourth One-Day International (ODI) against India at the Manuka Oval here on Wednesday:
Australia:
David Warner b Ishant 93
Aaron Finch c Ishant b Yadav 107
Mitchell Marsh c Kohli b Yadav 33
Steve Smith c Gurkeerat b Ishant 51
Glenn Maxwell c Pandey b Ishant 41
George Bailey c Rohit b Ishant 10
James Faulkner b Yadav 0
Matt Wade run out 0
John Hastings not out 0
Extras: (7lb,6w) 13
Total: (eight wickets; 50 overs) 348
Fall of wickets: 1-187 (Warner, 29.3 overs), 2-221 (Finch, 37.3), 3-288 (Marsh, 44.3), 4-298 (Smith, 45.4), 5-319 (Bailey, 47.6), 6-319 (Faulkner, 48.1), 7-321 (Wade, 48.3), 8-348 (Maxwell, 49.6)
Bowling:
Umesh Yadav 10-1-67-3
Bhuvneshwar Kumar 8-0-69-0
Ishant Sharma 10-0-77-4
Gurkeerat Singh Mann 3-0-24-0
Rishi Dhawan 9-0-53-0
Ravindra Jadeja 10-0-51-0
Match details:
Umpires: Richard Kettleborough (England), John Ward (Australia)
Television umpire: Paul Wilson (Australia)
Match referee: Jeff Crowe (New Zealand)
(IANS)