Having opened their campaign with losses to Australia and South Africa, a desperate New Zealand dominated the match after choosing to bat.
They made 227-9, boosted by half-centuries to captain Sophie Devine and Brooke Halliday.
Bangladesh never threatened the target, reduced to 33-6 at one point before being dismissed for 127 in the 40th over.
The result lifts the White Ferns to fifth on the standings in the eight-nation round-robin tournament heading into a match against winless Sri Lanka staring on Tuesday night (NZ time).
Devine said it was a relief to taste victory.
"Really pleasing for this side to finally get on the board, in tough circumstances.
"After the game against Australia, we felt we were good in places but the South Africa game was disappointing.
"This was important. Run-rate and with the weather around, every point is absolute gold. We have to focus on the next game. We cannot control what else happens."
Halliday top-scored with 69, while Devine's 63 was her third consecutive 50, consolidating her lead as the tournament's top run-scorer.
Devine said their fourth-wicket partnership of 112 "was not the prettiest" and the 36-year-old admitted she felt unwell at times as a type 1 diabetic.
"Diabetes was not playing ball. I was running low on sugar, so tried to get in as much sugar with coke and jelly beans," she said.
"It was just another challenge in conditions that are already tough enough."
Jess Kerr (3-21) skittled the Bangladesh top order, taking three of the first four wickets while fellow-seamer Lea Tahuhu (3-22) was just as lethal through the middle stages in her 100th one-day international.
