- Tag results for Australia Vs England
Buttler also shared an unbroken partnership of 113 runs with Chris Woakes for the seventh wicket to rebuild England's innings after the tourists had slumped to 189-6 in the 39th over.
England retained the same team that beat Australia by five wickets in Melbourne and then four wickets in Brisbane.
Finch blasted 106 from 114 balls, including nine fours and a six, to follow the 107 he made at the MCG on Sunday.
Australia have made three changes from the team that lost to England in Melbourne on Sunday including two debutants.
Finch's century was backed by fifties from Mitchell Marsh and Marcus Stoinis as Australia recovered from a shaky start as England got bogeyman Steve Smith cheaply.
Usman Khawaja batted for nearly nine hours over two days in amassing his 171, just missing his highest test score of 175.
The left-hander was unconquered on 166 off 368 balls and just off his highest Test innings of 174 scored against New Zealand in Brisbane in 2015.
At lunch, Australia were 277 for three and trailing the tourists by 69 runs with Khawaja unbeaten on 132 and Shaun Marsh not out two.
Australia had an early set back before half-centuries by David Warner and then Usman Khawaja helped place their team strongly in its push for a 4-0 series victory.
England's tail wagged with both Tom Curran and Stuart Broad flaying the Australian bowlers in an entertaining session, punctuated by some poor fielding by the hosts.
Test wicketkeeper Tim Paine replaces Matthew Wade in the 14-man squad for the five-match ODI series, starting from January 14.
The England captain wishes his own form was better on the field, and that his teammates' behaviour was better off it.
Players and commentators criticized the lifeless drop-in pitch used for the Boxing Day test between Australia and England.
Trailing 3-0 and the Ashes lost, England are considering a possible team change for the fifth Sydney Test with Moeen Ali's place under threat and Crane pushing for his first cap.
Starc missed the drawn fourth test at the Melbourne Cricket Ground because of a bruised right heel but is hoping to return for the New Year's Test.