Having seen off Zimbabwe last weekend, William Porterfield's side successfully defended their 164 for six against the Tigers.
A thrilling 71 from just 41 balls by opener Paul Stirling was the highlight of their innings.
Ed Joyce chipped in with a more considered 39 to boost the total further and, despite a knock of 52 from Shakib Al Hasan, Bangladesh were unable to reach their target.
The occasional spin of Kevin O'Brien (three for 24) eventually put the skids on their charge, which originally looked as though it would be in pursuit of a higher score when Ireland rattled off 40 runs in the first four overs.
Porterfield was the first man to go as he was run out with the score on 46, with Stirling and Joyce then sharing a second-wicket stand of 64 before the latter was castled by Elias Sunny.
Joyce carried on the charge before being snared by Mashrafe Mortaza, who finished with two for 33, as Ireland were unable to open their shoulders over the closing stages.
Bangladesh's response saw them lose Mohammad Ashraful early on for seven, before a quickfire stand of 63 between Tamim Iqbal and Shakib got them motoring.
But Tamim (35) offered a return catch to Alex Cusack and then Shakib lost his wicket to Nigel Jones as the wheels started to come off.
O'Brien then took three wickets in two overs to really put Ireland in command and, although Ziaur Rahman weighed in with a breezy 26, it was not enough and Bangladesh ended just short on 159 for nine.
Kamran Akmal blasted an unbeaten 92 from 50 balls as Pakistan beat India by five wickets with five balls remaining.
Virat Kohli (75) and Rohit Sharma (56) put on 127 for the fourth wicket and despite 4-23 from off-spinner Ravichandran Ashwin, Pakistan succesfully chased down 186.
South Africa beat New Zealand by nine runs as the Black Caps failed to get the 21 they needed from the final over.
Skipper AB de Villiers made 54 to lead the Proteas to 186-6 and despite 75 from opposition number Ross Taylor, NZ fell short with Dale Steyn picking up 4-26.
Chris Gayle hit four sixes in an unbeaten 65 as West Indies chased down 125 inside 16 overs to beat minnows Afghanistan by eight wickets.