Learn Finnish: Koira ei päästänyt kissaa pois. - The dog did not let the cat go away.

Because the action is negative. The dog did not let the cat go. In Finnish, whenever a sentence with an object is negative, that object must always be in the partitive case regardless of whether the action was completed.

Why does the verb use the päästänyt form instead of a regular past tense form like päästi?

This is the standard way to form the negative past tense. You use the conjugated negative verb ei combined with the past participle of the main verb (päästänyt). If it were an affirmative sentence, it would use the regular past tense: Koira päästi (The dog let).

I always confuse the verbs päästä and päästää. How can I remember which one to use?

Päästä is an intransitive verb meaning to manage to go, to reach, or to get to (the subject moves themselves). Päästää is a transitive verb meaning to let, to allow, or to release (the subject acts on an object). Because the dog is acting on the cat, we must use the transitive päästää.

The current lesson is about verbs of prevention like estää taking the -masta form. How does päästää fit into this?