The first match of the series was called off after Ireland cricketer Ruhan Pretorius had tested positive for Covid-19. But he returned negative in another test on Saturday and took part in Sunday's game.
Bangladesh won the toss and opted to field first at the Zahur Ahmed Chowdhury Stadium in Chattogram.
They restricted Ireland to 263 in 50 overs. Ruhan hit 90 for them off 125 balls, the highest score by any Ireland player in the match.
Also, James McCollum (41), Stephen Doheny (37), Harry Tector (31) and Shane Getkate (29) did well with the bat.
For Bangladesh, Sumon Khan and Rakibul Hasan took two wickets each while Mukidul Islam and Shafiqul Islam bagged one each.
In reply, Mahmudul (66) and Shamim (53*) came on strong and helped their team take a lead in the series.
Bangladesh needed to score 50 runs in the last five overs. Shamim hit two fours and as many as sixes, ensuring a win for his team.
Also, Saif Hassan (36), Yasir Ali (31) and Towhid Hridoy (31) also did well with the bat. Bangladesh scored 264 for six in 49.4 overs to seal the game by four wickets.
For Ireland, Ben White took two wickets while Josh Little, Ruhan and Shane Getkate took one wicket each.