There are a number of times when it is clear that Rainsford survives and the last few lines of the story suggest that, in the end, he lives and Zaroff dies.
First, Rainsford clearly survives after falling off the yacht. He swims toward the gun shots, correctly thinking they are coming from land.
With his remaining strength he dragged himself from the swirling waters.
During the first night he is being hunted, Rainsford manages to evade Zaroff enough that Zaroff decides to go home for the night. Although Zaroff could probably continue hunting, he is enjoying how cunning Rainsford is and wants to extend this hunting exhibition. Zaroff adds,
Your Burmese tiger pit has claimed one of my best dogs. Again you score. I think, Mr. Rainsford, I'll see what you can do against my whole pack. I'm going home for a rest now. Thank you for a most amusing evening.
Pursued again, Rainsford is forced to leap into the sea. He survives this as well. This is evident when Zaroff finds him waiting in his bedroom. The "game" is about to be completed. One man will kill the other. Zaroff says one will be food for the dogs and the other will sleep in the bed. The last line indicates that Rainsford kills Zaroff because it is he (Rainsford) who sleeps in the bed:
He never slept in a better bed, Rainsford decided.