I used to have the problem with memorizing a piece as well. You just need to listen to the song more, and play it even more. Once you get to the point where you can play a song relaxed, not focused on which notes to push, you know you've memorized it.
Memorizing a song doesn't necessarily mean you don't have to look at the notes. If you can play a song without looking at the notes, but you're very focused on which notes to push, you're only at the beginning of memorizing the piece. You need to learn the notes, then forget the notes and let the music guide you. If you don't believe me, try playing "Jingle Bells" or something easy. Chances are, you'll have no trouble memorizing it because you know the song so well and the music just flows through your fingers so you can listen to the music, and not be distracted by the correct notes to push.