Their 322-4 to overhaul Scotland's 318-8 with 11 balls to spare was the second highest successful run chase in World Cup history behind Ireland's 329 to beat England in 2011.
The win kept Bangladesh in the top four in Pool A along with New Zealand, Sri Lanka and Australia and they need to beat either England or New Zealand in their final two matches to be guaranteed a berth in the quarter-finals.
Opener Tamim Iqbal, who scored 95 off 100 deliveries to lead the run chase against Scotland, said their plan was to shut out thoughts of trying to overhaul a 300-plus total.
"So rather than thinking about the scoreboard, we were just enjoying the wicket. The wicket was fantastic and the outfield was really quick."
Captain Mashrafe Mortaza said the performance was better than when they scored more than 300 against Zimbabwe "because in the World Cup there is always pressure. So I think that is the best one".
Bangladesh, facing a gruelling travel schedule after playing Sri Lanka in Melbourne a week ago now head back to Australia to play England in Adelaide on Monday before returning to New Zealand to play the tournament co-hosts four days later in Hamilton.
Mortaza said they would wait to see the Adelaide wicket before deciding how many of their key spinners to include in their side to play England.
There would also be no immediate decision on opener Anamul Haque who suffered a serious shoulder strain diving to cut off a boundary against Scotland.
He took no part in Bangladesh's run chase and a team spokesman said they had been in consultation with doctors and could not immediately say how long it might take for him to recover.
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