Set 206 runs to chase, Bangladesh were all out for 167 with 22 balls to spare after they started terribly, losing their opening four wickets cheaply for 33 runs.
Shakib Al Hasan provided some resistance with 52 off 66 balls, but once his ninth one-day fifty ended after he edged Chigumbura behind the stumps, Bangladesh were never in contention.
Spinners Prosper Utseya and Raymond Price took two wickets each and the hosts conceded three run outs.
Earlier, asked to bat first, Zimbabwe appeared to be in trouble when they lost five wickets for 94 runs but Chigumbura hit 64 off 95 balls to give their innings some respectability.
He was run out when wicketkeeper Mushfiqur Rahim broke the stumps after collecting Roqibul Hassan’s throw from deep mid-wicket in the penultimate delivery of the final over.
Chigumbura had hit debutant pacer Mahbubul Alam for a six — the last of his five boundaries — earlier in the over to help Zimbabwe past the 200-run mark.
Alam was somewhat wayward, conceding 66 runs in his 10 overs, but all the other main Bangladesh bowlers maintained a tight line to keep Zimbabwe in check despite an aggressive start.
Left-armer Shakib was the pick, claiming 3-23, while off-spinner Naeem grabbed 3-32. Mashrafe Mortaza chipped in with 2-41.
Sri Lanka are the third team involved in the four-match event, which will end on January 16.