The dress shown is by designer Hannah Troy.
There is the copy from the ad: "New Talon Zephyr...fabulous nylon closure with the feel and flex of fabric. Borne on the high winds of fashion: A pouf of silk damask unburdened by the look and feel of hardware - accomplished by the new Talon Zephyr. A new nylon coil design makes it as light, soft and supple as fabric. And Zephyr's color lives forever, so it can't reveal that metal gleam along a seam. It's virtually snag proof. Magically self-healing. (If fabric or thread should catch in coils, bend it in half to open....remove fabric. Then zip it down, zip it up and it 'heals' itself!) Look for the tiny Zephyr in fall's most important fashions. Hannah Troy designed this one, and put special interest on the sleeves."
I'm not sure if Talon was the first company to introduce nylon coil zippers to the fashion industry, so I can't say that 1960 was the definitive year when the nylon zipper first appears in clothing. But I can say that Talon advertised it's new zipper in 1960 and that provides a valuable clue when trying to date some vintage garments.
UPDATE: Please note that the zipper I am referring to in this article is the nylon coil zipper, as shown on the right in this photo. Earlier plastic zippers were available and were most often used as a decorative element. They were heavier and looked similar to the one shown on the left in this photo. Please see my follow-up article on early plastic zippers from the 30's. Nylon wasn't invented until 1935 and it's use for zipper coils did not come until much later.