Yes, Apple does use its own payment system.
Using the Apple Store app, find the product you would like to purchase and add it to the shopping bag.
When done tap on the shopping bag tab at the bottom. You'll see a list of the products you have selected. Press the big black button named "Buy with (apple)Pay" (the button contains a graphical version of the Apple logo where I wrote (apple)).
Now you'll get a confirmation dialog and by confirming you can pay with Apple Pay.