Fast bowler Henry returned figures of 6-39 in 15.3 overs in the first innings as the Black Caps dismissed the home team for just 149 with fellow seamer Nathan Smith taking 3-20.
The tourists then responded with a total of 307 all out as opener Devon Conway top-scored with 88 while Daryl Mitchell struck 80 in the middle-order.
Zimbabwe fought well with the ball and paceman Blessing Muzarabani was the pick of the bowlers with figures of 3-73 in 26 overs while fellow seamer Tanaka Chivanga took 2-51 in 18 overs.
The home team were then poor with the bat again in the second innings as they could only post 165 all out to give them a lead of just seven runs.
Henry followed up his first day tally with 3-51 second time around while Will O'Rouke took 3-28 but it was spinner and stand-in captain Mitchell Santner that was the pick with 4-27 in 17.1 overs.
The second and final Test match of the series will again take place at the Queens Sports Club in Bulawayo starting on 7 August.