Gilchrist struck nine fours and five huge sixes off just 46 balls but the world champions were still made to work hard in chasing a target of 196 before going 1-0 up in the three-match series.
The hard-hitting left-hander dominated a 96-run opening stand with Simon Katich (18) in just 10.5 overs but Bangladesh, who had shocked the Aussies in a tri-series match in Cardiff last year, offered stiff resistence.
Gilchrist raced to his 45th half-century in just 27 balls to put Australia in control before a triple-strike by left-arm spinner Abdur Razzak made things difficult for them.
Gilchrist, who survived a chance on 39 when Syed Rasel failed to hold on to a catch in the deep off paceman Mashrafe Mortaza, was Razzak's first victim.
Razzak dismissed Gilchrist leg-before as the batsman shaped for a sweep and then trapped Andrew Symonds (zero) and captain Ricky Ponting (14) plumb with faster deliveries to finish with 3-36.
With Katich falling to Mortaza, Australia slipped from 96-0 to 133-4 and Bangladesh suddenly had them under pressure with the experienced Mohammad Rafique bowling a miserly spell from one end.
Michael Clarke (16) and Brad Hogg (seven) could not do much and in the end it took a watchful knock of 36 from Michael Hussey to anchor the team in the company of Brett Lee (12) with six overs to spare.
"I thought their spinners bowled very well to put us back a little. The ball expectedly turned a lot as the match progressed and that made things difficult for us," Ponting said.
Earlier, Aussie spinner Hogg bagged three wickets as Bangladesh were bowled out for 195. Hogg finished with 3-37 as the home side were dismissed in 47 overs after electing to bat.
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 Gilchrist when he charged down to Hogg.
Tushar Imran (20) and Mortaza (one) were Hogg's other victims as Bangladesh failed to keep their wickets intact.
Bangladesh were earlier troubled by pace bowlers Lee and Nathan Bracken, who picked up two wickets each to reduce the home side to 79-4 by the 19th over.
The other bowlers kept up the pressure and then Symonds came on to bag two wickets in his only over to end the innings.
"I thought we lost too many wickets against the new ball," Bashar said. "It is very hard to stay in the match when you lose so many early wickets but our spinners still did a great job."
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.