Chigumbura removed the home side’s top three batsmen, Tamim Iqbal for five, Junaid Siddique for nought and Mohammad Ashraful for 20, as Bangladesh were bowled out for 185.
Coming in at number seven, he then hit an unbeaten 60 from 50 balls, including nine fours and a six, as the tourists reached 189 for five inside 35 overs.
Only Mushfiqur Rahim, who made 56 from 77 balls, and Dolar Mahmud, who hit a rapid 41 from 30, managed more than 20 for Bangladesh, who were 62 for six at one stage.
Chigumbura finished with figures of three for 27 from eight overs, while Chamu Chibhabha and Kyle Jarvis took two wickets apiece.
Abdur Razzak, who claimed three victims, gave Bangladesh hope when he removed Brendan Taylor and Tatenda Taibu for ducks to leave Zimbabwe on 39 for three.
But, after Charles Coventry had been dismissed for 32, an unbeaten stand of 99 between Chigumbura and Stuart Matsikenyeri, who made 41 off 55, guided the tourists home with plenty to spare.