Cosgrove scored 74 and was associated in a 123-run opening stand with other left-hander Simon Katich (42 not out) as the visitors achieved their target of 125 in just 22.4 overs to complete a 3-0 drubbing.
Cosgrove, 21, adjusted to the slow pitch with ease to smash seven fours and two sixes during his 69-ball knock which ended at the brink of victory when he was bowled trying a big shot off spinner Abdur Razzak.
The South Australian became the seventh from his country to score a fifty on his One-day international debut, reaching the mark with a straight six off part-time spinner Rajin Saleh.
If Cosgrove played some impeccable shots straight down the ground, his partner Katich was superb square of the wicket during a 69-ball knock in which he struck five fours.
"It is wonderful to be playing for Australia and it feels awesome to have batted so well," said Cosgrove. "As I was stroking the ball well, I decided to go after their spinners, who had been bowling well in the series."
Cosgrove was given a chance as Australia decided to try their bench strength under Adam Gilchrist, who took over the reins from regular captain Ricky Ponting, who decided to rest himself.
"Cosgrove is a great talent and has impressed in domestic cricket. We have a good bunch of players waiting for their chances," said Gilchrist.
The reshuffled Australian team got to chase a small target after their bowlers took turns to torment Bangladesh's batsmen as the hosts were bowled out for a paltry 124.
Left-arm wrist-spinner Brad Hogg, who was pressed into the attack late, finished with three wickets, while Mitchell Johnson and Daniel Cullen bagged two scalps each.
Opener Rajin Saleh (37) and captain Habibul Bashar (33) were the only ones who came to terms with the slow pitch and Bangladesh lasted just 42.3 overs after electing to bat.
Saleh's painstaking knock came off 102 deliveries before he was bowled by left-arm spinner Michael Clarke while trying to sweep. Bashar was stumped by Gilchrist as he jumped out of his crease to Hogg.
Earlier, Bangladesh opener Shahriar Nafees (zero) fell leg-before on the third ball of the innings to left-armer Johnson.
Aftab Ahmed (one) and Mohammad Ashraful (13), who were both dropped for the previous match, gave up their wickets cheaply.
Ahmed flashed outside the off stump to be caught by Gilchrist off seamer Brett Dorey while Ashraful glided the ball straight to Daniel Cullen at short mid-wicket off James Hopes as Bangladesh fell to 41-3.
Saleh and Bashar added 39 for the fourth wicket but Bangladesh offered little else as Hogg, who had bagged three wickets each in the previous two matches, continued his fine run.
Australia won the preceding two-Test series 2-0.