Can you sacrifice a permanent with persist to itself if the sacrifice is the cost of an ability?

Absolutely yes. When a permanent has an activated ability that requires sacrificing itself as a cost (written as 'Sacrifice [this permanent]: ...' or similar), you may activate that ability and pay the sacrifice cost. The permanent is sacrificed as part of paying the cost, which causes it to go to the graveyard (CR 602.2a, 701.17).

Persist is a triggered ability that watches for the permanent going to the graveyard from the battlefield (CR 702.78a). It triggers when the creature dies — regardless of why it died. Dying as a cost to an activated ability still counts as dying, so persist will trigger as long as the creature had no -1/-1 counters on it when it died.

Once persist triggers and resolves, the creature returns to the battlefield from the graveyard under its owner's control with a -1/-1 counter on it. The activated ability the sacrifice paid for also resolves normally on the stack (the ability was already placed on the stack before the cost was paid, so it still resolves).

Example: You control Woodfall Primus (which has persist and 'Sacrifice Woodfall Primus: Destroy target noncreature permanent'). You activate the ability targeting an opponent's land. Woodfall Primus is sacrificed as cost, dies, and the ability resolves destroying the land. Then persist triggers and returns Woodfall Primus to the battlefield with a -1/-1 counter.

Key rules: paying costs happens in CR 602.2, sacrificing a permanent is defined in CR 701.17, and persist is covered in CR 702.78a. The creature's death from a cost payment is still a death that triggers persist.