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.
Saqibul Hasan's run out for three left them strugglng at 48-3.
It was left to Mohammad Ashraful (35) and Tamim Iqbal (29) to steady the innings through a 45-run stand for the fourth wicket but Iqbal played one shot too many and was bowled by Ireland skipper Trent Johnston.
Rankin returned for his second spell to remove a dangerous looking Mohammad Ashraful to end Bangladesh's hopes of a successful target chase. Ashraful hit five boundaries and a six during his 43-ball knock.
Skipper Habibul Bashar hung on for his 32 before Johnston bowled him to finish the match.
Earlier, Johnston's decision to bat first after winning the toss was justified by Porterfield who, in the company of Jeremy Bray (31), gave Ireland a confident start of 92.
Kevin O'Brien, who made 48 off just 44 balls with two sixes and as many fours, took Ireland past the 200-mark and gave the innings much-needed impetus during a 39-run fifth wicket stand with Johnston who made 30.
O'Brien added a solid 48 for the fourth wicket with Porterfield and was the key to Ireland ading 77 runs in the last ten overs before he was finally run out in the 48th over.
Porterfield, who has two hundreds to his credit in 15 one-day internationals, looked set for another three-figure mark before he paddle-swept paceman Mashrafe Mortaza to short fine-leg for Mohammad Rafique to take a simple catch.
He hit three boundaries during his steady 136-ball knock.
Bangladesh missed left-arm paceman Syed Rasel who sprained his ankle in a warm-up session, minutes before the start. Shahadat Hossain took his place but was unimpresive in his nine overs, conceding 51 runs.
Bray became the first of four run-outs in the innings.
Eoin Morgan (five) also fell to a run out, while Niall O'Brien (10) miscued a reverse sweep and was caught off Hasan to leave Ireland at 128-3.
Kevin O'Brien used his feet to the spinners, hitting Abdur Razzak for a towering six at long-on to ensure Ireland put a fighting total.
Ireland's last match is against Sri Lanka on April 18 in Grenada.