Disclosed herein is a golf ball having a non-ionomeric cover and method for making the same. The golf ball has been treated with crosslink-inducing irradiation at levels of at least 2 megarads up to 15 megarads. The non-ionomeric resin cover is made from material selected from the group consisting of non-ionomeric acid copolymers and terpolymers, polyamides, styrene block copolymers, polyamide block copolymers and syndiotactic resins. The golf ball cover is relatively soft, having a Shore D hardness that is no more than 10% greater than a cover having an identical composition but not radiation crosslinked. The cover is superior in cut resistance and scuff resistance to a conventional cover of substantially identical composition that is not covalently crosslinked. The golf ball cover of the invention is particularly useful for improving the durability of golf balls to be struck with sharp-grooved clubs.