The Scots had lost their last five Clydesdale Bank 40 matches but opener Berrington made exactly 100 to almost single handedly carry the minnows to 162 for seven.
Bangladesh, ranked fourth in the T20 world rankings, never got going in their chase and they were all out for 128, with Josh Davey and Majid Haq taking three wickets apiece.
With the scheduled one-day international against England later this month called off, this was Scotland's only match against Test-level opposition this summer and they were particularly grateful for Berrington.
The South African-born 25-year-old clubbed 10 fours and five sixes in his 56-ball knock at the top of the order after his side had opted to bat first. Calum MacLeod made the next highest score for Scotland with 19.
Mashrafe Mortaza took two for 22 and was easily the Tigers' best bowler, with Abul Hasan coming in for some heavy punishment, conceding 32 runs from his three overs, which included three consecutive fours off Berrington in the 13th over.
Bangladesh, who recorded a 3-0 T20 series victory over Ireland earlier this month, lost wickets at regular intervals to undermine their chase.
Opener Tamim Iqbal made a breezy 25 off 19 balls at the top of the order and Shakib Al Hasan top scored with 31 but their hopes vanished when he was seventh man out and Scotland duly closed out a convincing win, with Bangladesh all out in the 18th over.