The hosts, who had already secured the series with an unassailable 2-0 lead, won the toss and decided to bat at Zohur Ahmed Chowdury Stadium.
Ireland, though, made an early breakthrough when Mark Adair dismissed opener Litton Das for just five in the second over, with regular wickets following to leave Bangladesh struggling at 61 for seven.
Shamim Hossain (51) was the only batter to offer some resistance with a maiden T20 half-century as Ireland mopped up the innings with four balls to spare when Bangladesh were all out for 124.
Left-arm spinner Matthew Humphreys marked his T20 debut with a wicket from his first ball to bowl Rishad Hossain and then dismissed Taskin Ahmed for a duck in the same over to finish with two for 10 runs while Adair took three for 25.
In reply, after losing a couple of early wickets, Stirling soon set the tourists on their way with 77 off 41 balls, hitting 10 boundaries and four sixes, before he fell to legspinner Rishad, who was making his T20 debut.
A swift unbeaten 16 from Curtis Campher off nine deliveries and Harry Tector's 14 not out soon helped Ireland over the line at 126 for three with 36 balls to spare to secure a first victory in any format on tour in Bangladesh.