Hogg finished with 3-37 as the home side were dismissed in 47 overs after electing to bat on a good batting pitch on Sunday.
Left-arm wrist-spinner Hogg's scalps included the important one of captain Habibul Bashar, who struck a determined 52 and added 54 runs with Khaled Mashud (27) to rescue the struggling side.
Bashar struck three fours and a six over mid-wicket off debutant spinner Dan Cullen during his 11th half-century in 80 matches before he was stumped by Adam Gilchrist when he charged down to Hogg.
Tushar Imran (20) and Mashrafe Mortaza (1) were Hogg's other victims as Bangladesh failed to reproduce the magic that had helped them shock the Aussies in a tri-series one-dayer in Cardiff last year.
Bangladesh were earlier troubled by pace bowlers Brett Lee and Nathan Bracken, who picked up two scalps each to reduce the home side to 79-4 by the 19th over.
The other bowlers kept up the pressure and then Andrew Symonds came on to bag two wickets in his only over to end the innings.
Australia won both matches of a preceding Test series. They won the first match by a narrow three-wicket margin but thrashed Bangladesh by an innings and 80 runs in the next match.
The remaining one-dayers will be held in Fatullah on Wednesday and Friday.