Sharma thumped an incredible 16 sixes - the most in a one-day international innings - en route to only the third double-hundred in the format as India posted a mammoth 6-383.
The tourists looked to be heading for a heavy defeat when they slumped to 6-138 in reply, but Faulkner kept the game alive by striking the fastest ODI century by an Australian.
His maiden ton came from 57 balls and set up an unexpectedly tense finale in Bangalore.
Faulkner was eventually dismissed for 116 to end Australia's innings at 326 in the 46th over, giving India victory by 57 runs and a 3-2 series success.
Glenn Maxwell and Shane Watson also impressed with the bat for Australia, striking 60 and 49 respectively from 22 balls apiece.
However, the latter was clearly inconvenienced as he batted at number eight having suffered a hamstring problem during India's innings, which ended with 115 runs coming from the final six overs.
Sharma moved from 100 to 200 in the space of 42 deliveries and there was a similarly swift contribution from India skipper, MS Dhoni, who cracked 62 from 38.
On another tough day for the bowlers, India's Vinay Kumar finished with the most abject analysis, returning 1-102 from nine overs.
Clint McKay (1-89), Nathan Coulter-Nile (0-80) and Faulkner (1-75) were others to face severe punishment in a match featuring 38 sixes - another record.
In addition to his individual record haul of maximums, Sharma registered 12 fours in a wonderful 158-ball knock. Only fellow countryman Virender Sehwag, who hit 219 against South Africa in 2011, has managed a higher 50-over score.
Shikhar Dhawan contributed 60 to an opening partnership of 112 before falling lbw to Xavier Doherty (2-74) and Virat Kohli was then run out for a duck.
Suresh Raina and Yuvraj Singh were unable to make much of an impression, but it mattered little as Sharma, who drove effortlessly throughout, took centre stage.
The loss of early wickets looked certain to undermine Australia's reply, but Maxwell and Watson cleared the ropes with regularity in thrilling cameos.
Faulkner offered even greater resistance, plundering six sixes and sharing 115 for the ninth wicket with Clint McKay (18).
With six overs remaining, an Australia win was still a possibility, but Ravindra Jadeja (3-73) then bowled McKay and Mohammad Shami (3-52) had Faulkner caught in the deep to bring proceedings to an end.