The Afghans now lead the series 2-1 with two matches remaining, as the teams warm up for the 2019 World Cup qualifiers in Zimbabwe next month.
Zimbabwe won the toss and batted first, but collapsed from 117-3 to 154 all out as 19-year-old Khan finished with figures of 5-24.
Rahmat Shah and Nasir Jamal both struck half-centuries in Afghanistan’s reply as they wrapped up victory with 22.3 overs remaining.
Zimbabwe recovered from 44-3 with Sikandar Raza hitting 38 and Craig Ervine 39, but both fell LBW to Khan as the teenager, who was bought by Sunrisers Hyderabad for $1.41 million in last month’s IPL auction, ran through the middle order.
Mujeeb Ur Rahman, 16, again impressed opening the bowling with his leg-breaks and claimed the key wicket of second-match centurion Brendan Taylor on his way to 3-45.
Big-hitting Afghanistan wicketkeeper Mohammad Shahzad — continuing his return from a drugs ban — fell early, but Shah and Jamal combined to put on 69, before Mohammad Nabi and Najibullah Zadran finished the job with some lusty blows.
Afghanistan will look to win the series in the fourth ODI at Sharjah on Friday, with the sides set to meet again in Group B of World Cup qualifying on March 6.
Qualification for the 10-team World Cup in England and Wales next year would continue Afghanistan’s remarkable rise, with the war-torn nation to play their maiden Test match against India at Bangalore in June.