What wins out in the end: instinct or reason? How does Rainsford survive?

Actually, the answer would be both. Rainsford escapes Zaroff by jumping from the cliff. His jump wasn't a planned, he didn't think about it rationally. He simply followed his instinct.... if he hadn't jumped he would have died, so he took his chances that he wouldn't die on the rocks. After he survives the jump, Rainsford used rational thought to plan his appearance in Zaroff's room. Thus, he uses instinct and reason.