Devine, who scored 112 and 85 in losing causes against Australia and South Africa, this time scored 63 off 85 balls on Friday to help rescue the White Ferns from another shaky start.
Brooke Halliday top-scored with 69 off 104 balls and took the player-of-the-match award as New Zealand recovered from 3-38 to post 9-227 in tough hot, humid conditions.
When they came together, the 36-year-old skipper Devine and Halliday had to battle hard, putting on 112 for the fourth wicket to extricate NZ from their tricky spot.
Devine smiled that it hadn't been her "prettiest" innings and she'd been running low on energy levels, but added: "It's really pleasing for this side to finally get on the board, in tough circumstances."
Halliday added: "Our plan was clear – we wanted to be boring, looking for ones and twos. We wanted to keep things simple with bat and ball. We knew if we did that, we will fare well."
In reply, Bangladesh were bundled out for 127 in 39.5 overs with medium pacers Rosemary Mair, Jess Kerr and Lea Tahuhu sharing eight wickets between them and only Fahima Khatun showing much resistance for Bangladesh with her 34.
Opting to bat, New Zealand lost three wickets in 11 overs as wrist spinner Rabeya Khan continued her good form by having Georgia Plimmer stumped for four and then bowled Amelia Kerr for just one.
Suzie Bates was then run out for 29 before Devine and Halliday righted the ship with their 165-ball stand. Halliday made her 50 off 80 balls, and Devine reached her half-century off 82.
They were the first New Zealand pair to have a 50-plus stand in three consecutive games, and it was also the White Ferns' second-best partnership for the fourth-wicket or lower in ODIs.
Halliday fell in the 39th over when she was caught off Fahima Khatun, while Devine was bowled in the 44th over. Maddy Green's 25 off 28 balls with three fours held the lower-order together, allowing them to cross 220.
In reply, Bangladesh's batters collapsed in a heap as New Zealand's medium pacers reduced them to 7-66. None of the top five batters crossed five runs.
Rosemary Mair – playing her first game of the tournament – picked 2-20 in six overs. Kerr took 3-21 in eight overs, while Tahuhu returned 3-22 in six overs.
New Zealand's first win moved them up one spot to sixth in the standings, before they play co-hosts Sri Lanka in Colombo on Tuesday.
Bangladesh stayed fifth in the table with a second loss in three games, and will next play South Africa in Visakhapatnam.
Australia still top the table on five points from their three matches, but also unbeaten England could go top on six points should they beat their hosts Sri Lanka in Colombo on Saturday.
