Marsh plundered an unbeaten 177 from 132 balls on Saturday in Pune, helping Australia pull off their biggest ever World Cup chase as they hauled in Bangladesh's 8-306 with 32 balls to spare.
The win ensures Australia will enter Thursday's semi-final against South Africa as one of the competition's form teams, rebounding from two opening losses to India and the Proteas.
Steve Smith and Mitch Marsh chat after their match-winning partnership in Pune.
Marsh's performance will please Australia most.
The West Australian smashed 121 opening against Pakistan earlier in the tournament, but had recorded scores of nine, 36 and 24 since being forced down to No.3 on Travis Head's return.
But on Saturday afternoon, this was classic Marsh.
He drove with the ball great power, putting the pressure back on all of Bangladesh's bowlers and punishing them whenever they strayed on being marginally too full.
Seven of his nine sixes came on the drive, while he also hit 10 of his 17 fours in the same manner.
Marsh was on from the outset, scoring 26 runs from boundaries in his first 11 balls before reaching 50 in 37 deliveries and a century in 87.
After Head went early for 10 and David Warner hit 53, Seve Smith also struck a timely 63 not out ahead of big selection call for the semi-finals.
Smith looked calm and composed in his 64 balls at the crease, never appearing troubled as he accumulated runs and played one heaving slog sweep for six off spinner Nasum Ahmed.
Smith's runs came in an unbeaten 175-run third-wicket stand with Marsh, as Australia consider the make-up of their middle order to face South Africa.
It also meant Marnus Labuschagne did not have a chance to press his claims with the bat, after pushing the case with a superb morning in the field.
Considered the man most likely to miss out in a squeeze between he, Smith and Marcus Stoinis, Labuschagne effected two smart run outs in Bangladesh's innings.
The 29-year-old first removed Najmul Hossain Shanto on 45 when he slid to haul a ball in at square leg and threw in one motion to catch the Bangladesh captain short.
Then as Bangladesh were gaining momentum, Labuschagne charged in, leapt forward and threw the ball in mid-air to have Mahmudullah (32) with a direct hit.
Marnus Labuschagne was dynamic in the field, completing two run outs during the Bangladesh innings.
Originally left out of Australia's World Cup squad, Labuschagne has now played 17 straight matches since being called into the ODI squad to tour South Africa.
And he has continually proven an asset in the field.
Labuschagne has averaged 35.75 to Smith's 38.28 in India, while Smith is striking at 88.74 to Labuschagne's 77.08.
Adam Zampa (2-32) also continued his terrific tournament, taking the key wicket of Liton Das (36) when he deceived him in flight and had the right-hander caught at long off.
He also had Mushfiqur Rahim caught at midwicket for 21, while Josh Hazlewood (0-21 from seven overs) also bowled well.
Sean Abbott took 2-61 in his first World Cup match after being called in for the rested Mitchell Starc, who will return for the semi-final.
Towhid Hridoy did well to anchor Bangladesh's innings with 74 off 79, but when he was out trying to take on Stoinis (1-45 from five overs) Australia were able to contain late.
