A common misconception in Magic is that you can prevent an activated or triggered ability from being resolving by destroying the source of the ability. This is not true and has never been the way that Magic works. Once an ability is played or triggered and put on the stack it is independent of its source. Removing the source from the battlefield will not counter the ability.

For example, if you are targeted by a the ability of a Goblin Fireslinger responding by targeting the Fireslinger with Shock will not prevent you from receiving damage. Yes, the Fireslinger will not be in play when its ability resolves, but that does not affect the ability. As the Fireslinger is no longer in play the game will use the last known information about the Fireslinger to determine the characterictics of the source if needed.

Similarly, if a creature enchanted by a Brink of Disaster becomes tapped destroying the Brink of Disaster before this trigger resolves will not save this creature.