Rashid was signed by Sunrisers Hyderabad for $1.41 million in last month's Indian Premier League auction and returned figures of 3-19 from his four overs as Zimbabwe were restricted to just 120-9.
Mohammad Shahzad clubbed a pair of sixes in an 11-ball innings of 20 to ensure Afghanistan, who will play their maiden Test match in June against India, made a promising start to their chase.
A middle-order wobble reduced Afghanistan to 85-5 but Nabi held firm to guide his team to victory with more than five overs to spare.
The second and final match of the T20I series takes place at the same ground on Tuesday before the sides face off in five ODIs, starting on February 9.
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