Ireland made 134-8 off their 20 overs, Isobel Joyce scoring 41 and Jahanara Alam taking 5-28 at YMCA Sports Club.
Bangladesh scored 135-6 in reply, scrambling a run from the last ball to take a thrilling four-wicket victory.
The teams will meet again at Malahide on Friday, with the concluding match being staged at Pembroke on Sunday.
Despite losing their two openers – Clare Shillington and Cecelia Joyce – early, the Irish middle order rallied to put on a positive batting display.
Gaby Lewis was an early aggressor, striking 28 runs from 16 balls, including a massive six and a number of powerful well-timed boundaries.
Lewis was supported by Ireland captain Laura Delany (22 runs from 22 balls), before Joyce entered the fray and built an anchor innings off 41 balls.
Kim Garth played a cameo innings of 20 runs from 19 balls.
Bangladesh’s response got off to a rocky start, with opener Shamima Sultana being dismissed in the second over for four.
Several stabilising partnerships followed, built around the powerful batting of Nigar Sultana (46 off 41 balls), who eventually fell to a fantastic low-down diving outfield catch by Garth.
With three overs remaining Bangladesh required 27 runs to win, but the visitors took 23 runs off overs 18 and 19 to put themselves in the driving seat.
Eimear Richardson stepped up to bowl a tense final over, with three runs coming from the first four balls.
Then, on the second last ball, a desperate single was chanced by batter Sanjida Islam, but a spectacular pick up and throw from Cecelia Joyce at cover hit the stumps directly and ran Islam out.
A run off the last ball proved enough for Bangladesh however.