Although debutant Karun Nair fell early in the Indian chase, KL Rahul and Ambati Rayudu made light work of the target as they forged an unbroken stand of 162.
Rahul finished the game off in style as he launched into the stands to end on 100 not out, while Rayudu was unbeaten on 62.
Earlier, Bumrah, who was playing his second ODI, recorded figures of 4 for 28 as the Zimbabweans were bowled out for 168 after India had won the toss and asked the home side to bat.
Opening bowlers Barinder Sran and Dhawal Kulkarni both used the early-morning swing on offer to make inroads at the top of the Zimbabwean innings, and that damage was compounded when Bumrah dismissed opener Chamu Chibhabha and then Vusi Sibanda in his opening spell.
With the hosts reduced to 69 for four in the 20th over, spinners Axar Patel and Yuzvendra Chahal were able to strangle Zimbabwe through the middle overs.
Patel and Chahal picked up a wicket apiece and conceded just 53 runs in their combined 20 overs as Zimbabwe limped along.
Top scorer Elton Chigumbura attempted to lift them to a respectable score with a late flourish, but was eventually bowled by Bumrah for 41 in the final over of the innings.
The second game in the three-match series takes place at the same venue on Monday.
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