Yes! You need a specific facial moisturizer. And I promise I'm not just making you spend more money, there really are a couple of reasons why you shouldn't use your body moisturizer on your face.
You face is more sensitive than your body, and it has special needs that the rest of your body doesn't have. Some people are acne prone, some are prone to fine lines and wrinkles. A body moisturizer doesn't need to worry about doing anything other than hydrating your body. Many times there are added fragrances included in body moisturizers to give you a soft scent as well as hydration, and 99% of the time there is no SPF sun protection. When it comes to a facial moisturizer, you need to make sure you're picking one that is meant for your skin type. Here are the four skin type characteristics for you to see where you fit.
If you just use your body moisturizer, you could wind up with more frequent breakouts and most likely oily skin.
The second reason to stick with a facial moisturizer is for sun protection. Many facial moisturizers add in SPF coverage these days. If your facial moisturizer doesn't have it, make sure you apply before you head out.
So while it might be an extra step in your morning routine, I think you'll be happier with the condition of your skin after you start using a moisturizer meant specifically for your skin type, and your face.