Can Sacred Ground mitigate Annihilator?
Yes, this works as you describe.

Although Sacred Ground has no rulings in Gatherer; we can look at a similarly-worded card, such as Nephalia Academy.


  If a spell or ability an opponent controls causes you to discard a card, you may reveal that card and put it on top of your library instead of putting it anywhere else.


This has the following ruling:


  If a spell or ability an opponent controls causes you to make a choice and you choose to discard a card, Nephalia Academy’s ability will apply.


This should make it clear that even when you have a choice to make; if you choose the thing that the ability cares about, it counts as your opponent's spell or ability "causing" it to happen.

In the case of Annihilator; your opponent controls the Annihilator triggered ability which is what causes you to sacrifice a permanent.

This would be no different if you controlled an Eldrazi Conscription enchanting an opponent's creature; because your opponent's creature is still the object with the Annihilator ability. It would be different if Eldrazi Conscription said "whenever enchanted creature attacks; defending player must sacrifice 2 permanents". If that were the wording; then you would be the controller of the ability, not your opponent.