Chasing 185 for seven, Kent were already behind the asking rate at the midway point.
But there was still time for Fabian Cowdrey, the grandson of former Kent and England skipper Colin and son of Chris Cowdrey, to make his first-team debut where he finished unbeaten on 20.
Kent's on-loan Durham seamer Mitchell Claydon earlier took five for 31 when the Kiwis batted.
But Martin Guptill (56) and Ross Taylor (52) made the crucial runs to put New Zealand in charge.