When you cast a spell, you first put it on the stack, then choose targets (if the spell requires any), and then you pay its casting cost.
Finks' Persist ability triggers when you sacrifice it, and then goes to the stack. Eldritch Evolution is already on the stack at this point, so the persist trigger will enter the stack above it, and therefore will resolve first.
Before. In general, if a triggered ability triggers as you pay the cost of a spell or ability (such as sacrificing a creature to pay for Eldritch Evolution), that triggered ability will go on the stack above that spell or ability (C.R. 405.2) and get to resolve before it (C.R. 116.4).
Finks' Persist ability triggers when you sacrifice it, and then goes to the stack. Eldritch Evolution is already on the stack at this point, so the persist trigger will enter the stack above it, and therefore will resolve first.
EDIT (Nov. 1): Add rule citations.