It's definitely OK to get your morning dose of sunscreen from your moisturizer! That's what makes it so easy to remember to apply facial sunscreen. You're doing the right thing, and you don't need to add another step or product in.
But know that the sun protection won't last all day so if you're outdoors, you'll need to apply some later in the day. You could use more of your moisturizer, but if it's too expensive or you don't want to use it for any reason, a facial sunscreen of any kind will do.