Set a target of 190 to win, the visitors got there without the loss of a wicket in 30.5 overs as Dhawan scored 81 from 113 balls and Gill a more brutal 82 from 72 deliveries.
Zimbabwe used eight bowlers in their attempt to restrict their opponents, but there was no reward against the India openers.
The visitors had elected to bowl first and were quickly into the wickets as they reduced the home side to 66 for five.
Seamer Deepak Chahar (30) removed both openers and finished with figures of 3-27, well supported by the more expensive Prasidh Krishna (26), who took 3-50 in eight overs.
The guile of left-arm spinner Axar Patel (28) was also impressive as he removed innings top-scorer Regis Chakabva (35) and finish with figures of 3-24.
A 70-run ninth-wicket stand between Brad Evans (25) and Richard Ngarava (24), who top scored with 34, added some respectability to the home score, but it never looked like a target to trouble India.
The second ODI will be played on Saturday and the third on Monday, both also in Harare.
