Ireland's impressive total of 243-7 was built around a steady 85 by William Porterfield and 48 by Kevin O'Brien before bowling Bangladesh out for 169.
The win assured Ireland a 12th place in the ICC (International Cricket Council) one-day rankings as they have now met the condition of beating two Test playing nations.
They beat Pakistan by three-wickets in the preliminary round to qualify for the Super Eights in their maiden World Cup appearance.
The defeat proved an anti-climax to Bangladesh's campaign in which they beat India in the group stages and upset South Africa in the second round.
Their last match is against the West Indies on Thursday.
Bangladesh, who had an outside chance of a semi-final place, spurned that with some uninspired bowling and unimaginative batting.
Irish spearhead Boyd Rankin, living up to his habit of taking early wickets in the tournament, had Shahriar Nafees (seven) caught behind and then Andre Botha had Aftab Ahmed (12) in the same manner.