England captain Alastair Cook dropped two catches at first slip in the first six overs of the day, giving chances to Rogers on 19 and David Warner on 22.
Paceman Stuart Broad reacted with hands on hips after wicketkeeper Jonny Bairstow failed to attempt a catch from Rogers and Cook moved late.
Cook also dropped a sitter at waist height off Ben Stokes' bowling.
Warner added three more runs before his attempted upper-cut off Stokes was caught by Bairstow at 1-64.
Bairstow messed up again in the first over after lunch by failing to hold a low chance off the bowling of Broad with Rogers on 81.
Rogers notched his second Test hundred, the adopted Victorian reaching the milestone on his home ground at the MCG.
The left-hander brought up his 135-ball hundred with a stylish drive to the cover boundary, scoring more freely than most batsmen in this low-scoring Test.
The 36-year-old raised both arms in a salute to the crowd of 38,522.
Rogers became Australia's sixth century-maker of the series, compared to England's one.
Spinner Monty Panesar finally broke the second-wicket partnership when Rogers was caught behind for 116 at 2-200 after hitting 13 boundaries.
It was the first time Australia had successfully chased a target of more than 200 in an MCG Test since 1960-61 against West Indies.
Australia were bowled out for 204 on Saturday in reply to England's first innings of 255.
England collapsed in their second innings to be all out for 179.
More than 270,000 fans attended the match including a world-record crowd of 91,092 on Boxing Day.
Australia will assess the fitness of Watson (groin) and pace bowler Ryan Harris (knee) for the fifth and final Test starting in Sydney on January 3.
Clarke is seeking to join Ricky Ponting (2006-07) and Warwick Armstrong (1920-21) as captains who have led Australia to a five-nil Ashes sweep.
Clarke said grabbing a 4-0 lead was an amazing feeling that he couldn't have predicted.
"Probably not, to be perfectly honest,'' he said.
"To be 4-0 up, it's hard to describe.
"There were some great individual performances.
"Our goal is to win 5-0. I dont' see any reason why not.''
Clarke praised Rogers and his bowling attack, led by Mitchell Johnson and spinner Nathan Lyon who both claimed five-wicket hauls.
Johnson claimed his third man-of-the-match award for the series and lifted his wicket-taking tally to 31 victims after grabbing 5-63 and 3-25.
"I thought that Chris Rogers probably deserved it more than me,'' Johnson said.
Cook said his struggling side were trying as hard as they can.
"We've got to dust ourselves off, get over it tonight and head up to Sydney,'' he said.
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