Hello,
I am not sure if baby acne is considered the same as Eczema. My son had eczema and his pediatrician told me to use hyrdocortisone 1%. It started around 2 months old, and lasted for 6 months or more. I used it twice a day, it took a while for it to clear up completely. As it was healing his skin tone where he had the eczema was lighter than the rest of his face. After applying the hyrdocortisone, I would also use an over the counter moisturizing cream. I think vaseline would work too. Also, I use Dove soap for baths. Not sure if this will help you, but it worked for me.