Can you sacrifice a creature in response to it being targeted by exile removal?

When an opponent casts an exile removal spell (such as Swords to Plowshares), it goes on the stack and both players receive priority before it resolves. This window allows you to activate a sacrifice outlet — an ability that lets you sacrifice the creature as a cost — in response, removing the creature from the battlefield before the removal spell resolves (CR 117.1, CR 116.2b).

When the exile removal spell then tries to resolve, it checks for its target. Since the creature is no longer on the battlefield, the target is no longer legal. A spell with no legal targets is countered by the rules, so it simply fails to resolve and goes to the graveyard (CR 608.2b). The creature avoids exile entirely.

This is especially relevant against "exile when it dies" replacement effects or to trigger "when this creature dies" abilities (like a Sacrifice: draw a card outlet), since the creature goes to the graveyard via sacrifice rather than exile.

Concrete example: Your opponent casts Swords to Plowshares targeting your Gravecrawler. In response, you activate Viscera Seer, sacrificing Gravecrawler. When Swords to Plowshares goes to resolve, its target is gone — it is countered by game rules, and Gravecrawler goes to your graveyard instead of being exiled.

Note: this only works with activated sacrifice outlets (abilities with a sacrifice cost you can activate at instant speed). You cannot use a sorcery-speed effect or a triggered ability that hasn't triggered yet to sacrifice in response (CR 509.2, CR 602.2).