In the past, Japanese ages used to be counted similarly to Korean ages, as ajsmart's comment talks about, but Japanese ages have been "westernized" for a long time now.
My Japanese friends who have started families always refer to their children's age from birth. I have heard newborn babies' ages described as []{} "zerosai", and also []{} "ikkagetsu", and so on for months, []{} "isshuu", and so on for number of weeks, and if you want to specify down to the day for some reason, you add …[]{} "to ichinichi" etc for days.
So a baby that is 3 months old would be []{} "sankagetsu".
A baby that you want to describe as (for some strange reason) 2 years, 4 months and 5 days old would be []{}[]{}[]{} "ni sai, yonkagetsu to itsuka".