Marsh's second hundred of the World Cup ensured Australia, already guaranteed a last-four clash with South Africa, made light of their target of 307 to win with 32 balls remaining.
This was just Marsh's third hundred in 87 one-day internationals and surpassed his 121 against Pakistan earlier in the tournament as his highest score.
The all-rounder, strong on the drive and square of the wicket, faced 132 balls including 17 fours and nine sixes.
Marsh also shared a stand of 120 with David Warner (53) and put on an unbroken 175 with Steve Smith, the star batsman hitting the winning boundary in a morale-boosting 63 not out after returning from vertigo.
Victory meant Australia ended the group stage with a seventh successive win after their campaign for a record-extending sixth men's World Cup title started with two defeats.
Bangladesh's 306-8, their highest total of the tournament, left Australia needing to surpass their record World Cup-winning chase of 292 against Afghanistan in Mumbai on Tuesday - a match where Glenn Maxwell's stunning unbeaten 201 sealed an astounding three-wicket win after they had collapsed to 91-7.
Maxwell was rested on Saturday, but Australia had no need of his heroics as they overpowered a Bangladesh attack missing captain Shakib Al Hasan after the all-rounder was ruled out with a broken finger.
Nasum Ahmed's 10 wicketless overs cost 85 runs, with this loss leaving Bangladesh dependent on other results for a top-eight finish that would qualify them for the 2025 Champions Trophy in Pakistan.
Australia lost Travis Head (10) early in their chase when he played on trying to slog Taskin Ahmed.
Warner plundered runs on an excellent batting pitch before the left-handed opener, following a 52-ball fifty, surprisingly chipped Mustafizur Rahman to stand-in skipper Najmul Hossain Shanto at mid-off.
The powerful Marsh however, pressed on, pounding spinner Mehid Hasan Miraz for a six and a four before suffering cramp, although nothing as severe as the attack that afflicted Maxwell against Afghanistan.
Earlier, Towhid Hridoy's 74, his maiden World Cup fifty, was the cornerstone of a Bangladesh innings were several batsmen gave away their wickets.
Adam Zampa was the pick of Australia's attack with 2-32 as the leg-spinner became the leading bowler in the tournament with 22 wickets, while Marnus Labuschagne derailed the innings by expertly running out the well-set Najmul (45) and Mahmudullah (32).
Hridoy's 79-ball innings ended when he holed out off Marcus Stoinis, his exit sparking a flurry of wickets.
