After the Irish won the toss and chose to bowl first, Max Sorensen (3-31) and John Mooney (3-32) took three wickets apiece as Bangladesh were dismissed for 189 in 48.2 overs.
Former England top order batsman Ed Joyce hit 47 to help set the Irish on their way before Andrew Balbirnie (63 not out) and Kevin O'Brien (23) teamed up for a sixth-wicket partnership of 41 to take them to the brink of victory.
After O'Brien was caught behind off Taskin Ahmed, Mooney returned with his bat to add two runs and help Balbirnie steer the side past their target with just over four overs to spare.
Ireland have produced shocks at both their previous World Cups, tying with Zimbabwe and beating Pakistan in 2007 and stunning England on the back of O'Brien's 50-ball century four years later in India.
The victory will be a huge confidence boost for the Irish, who were well-beaten by fellow Celtic minnows Scotland in their first warm-up game on Tuesday.
They will certainly now fancy their chances of a victory in their tournament opener against an out-of-sorts West Indies side in Nelson, New Zealand on Monday.
Bangladesh, who open their campaign against Afghanistan next Wednesday, never really managed to get their innings going and Soumya Sarkar was their top scorer with 45 from 51 balls.
Left-arm spinner Taijul Islam was the pick of their bowlers with two for 29 including the dismissal of Joyce, but an Irish victory never really looked in doubt.
(Reporting by Nick Mulvenney, editing by Amlan Chakraborty)
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