I have since found an explanation on another discussion on the same sentence. It's because 'eerste' is always spelt with the 'e' on the end.
Ah, but 'eerste' doesn't belong to the adjective rule you are referring to. It's a number-like thing:
- Eerste - first
- Tweede - second
- Derde - third
- Vierde - fourth
They all end on an 'e'. ;)
We also have the word 'eerst'; which indicates a place in time. As in 'first I am going to do this, and then I'll do the rest'. That kind of 'first'.
- Eerst ga ik naar school, daarna ga ik winkelen. - First I'm going to school, afterwards I'll go shopping.
- We gaan eerst. - We go first.
- De eerste plaats - The first place.
It is specific: there's a possessive determiner (haar). So, haar eerste kind