The medium-pacer picked up three for 27 as Bangladesh were bowled out for 186 in 46.5 overs. Chigumbura then hammered an unbeaten 60 in 50 balls, with nine fours and one six, to guide Zimbabwe to a comfortable win with 15.2 overs to spare.
He and Stuart Matsikenyeri (47 not out) shared an unbroken sixth-wicket stand of 99.
Left-arm spinner Abdur Razzak claimed three wickets in his first over as Zimbabwe wobbled at 39 for three. He eventually finished with three for 36.
Earlier, vice-captain Mushfiqur Rahim scored a sedate 56 off 78 balls after Bangladesh had slid to 62 for six on a bouncy track.
Dolar Mahmud chipped in with 41 in 30 deliveries, including three sixes and four fours, as he and Mushfiqur put on 54 for the ninth wicket.
Chigumbura drew first blood in the second over of the match, having Junaid Siddique dismissed for a duck, and followed it up with the wicket of Tamim Iqbal, caught behind by Tatenda Taibu. As Joseph Chibhabha too got into the act, Bangladesh lost their middle-order cheaply and looked in danger of being bowled out for under a hundred before Rahim bailed them out.
With Naeem Islam for company, Rahim put on 53 for the seventh to push the total to three-figures, then the ninth-wicket pair gave the bowlers something to bowl with. The second game in the five-match series is also in Dhaka on Thursday.
Score Board
BANGLADESH
Iqbal c Taibu b Chigumbura 5
Siddique c M’enyeri b Chigumbura 0
Ashraful c Taibu b Chigumbura 20
Raqibul (run out) 16
Shakib c Taibu b Chibhabha 13
Rahim c Taibu b Jarvis5 6
Mahmudullah c Price b Chibhabha 4
Islam c Coventry b Price 21
Razzak c Utseya b Cremer 3
Mahmud c Masakadza b Jarvis 41
Hossain (not out) 0
Extras (LB-5, W-2) 7
Total (all out, 46.5 overs) 186
Fall of wickets: 1-0, 2-5, 3-41, 4-56, 5-58, 6-62, 7-115, 8-124, 9-178.
Bowling: Jarvis 8.5-1-43-2, Chigumbura 8-1-27-3, Chibhabha 8-0-28-2, Utseya 4-1-15-0, Price 9-0-31-1, Cremer 9-0-37-1.
ZIMBABWE
Masakadza c Hasan b Hossain 10
Chibhabha lbw Razzak 26
Taylor lbw Razzak 0
Taibu b Razzak 0
Coventry c Razzak b Mahmud 32
Matsikenyeri (not out) 47
Chigumbura (not out) 60
Extras (LB-1, W-10, NB-3) 14
Total (for 5 wkts, 34.4 overs) 189
Fall of wickets: 1-39, 2-39, 3-39, 4-54, 5-90.
Bowling: Mahmud 5-0-26-1, Hossain 8-0-59-1, Shakib 10-0-49-0, Razzak 8.4-1-36-3, Mahmudullah 1-0-6-0, Islam 1-0-7-0, Ashraful 1-0-5-0.
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