There are two definitions of "prevail" that would apply here. Prevail can mean to persuade. So, the first instance of Rainsford's attempt to persuade the General occurs when they are eating dinner. Rainsford is appalled to learn that Zaroff hunts human beings. Rainsford claims this is murder and therefore different from killing animals. Zaroff replies that he can not believe Rainsford would be that idealistic and naive about human life. Rainsford adds that he is "a hunter, not a murderer."
Zaroff tries to persuade (prevail upon) Rainsford that murder is an unnecessary and inappropriate concept to use. In other words, for Zaroff, killing is killing. He says that there are two kinds of people: strong and weak. He adds, "The weak of the world were put here to give the strong pleasure. I am strong. Why should I not use my gift?" Rainsford is still appalled and this is ironic because he expressed a similar notion regarding hunting animals while he was still on the ship. He had said that there are hunters and "huntees."
At the end of their conversation, Zaroff is not persuaded by Rainsford. His next attempt to prevail upon the general is when he argues that he should be able to leave the island. Zaroff gives him the option of becoming the hunted or facing Ivan. Reluctantly, Rainsford takes his chances as Zaroff's prey.
Prevail also means to prove something or someone is superior. Rainsford's quest to stay alive is the next effort to prevail. During the first night, Rainsford tries to lure the General down a circular path. He is unsuccessful and Rainsford realizes that the General is simply pretending he can not find him in order to hunt him another day. The General prevails on this first night. But in the end, Rainsford prevails by jumping into the sea.