While it is true that author Richard Connell never specifically declares that Rainsford has won the game, the reader can assume that he is the lone survivor when he settles down in Zaroff's bed.
The general made one of his deepest bows. "I see," he said. "Splendid! One of us is to furnish a repast for the hounds. The other will sleep in this very excellent bed. On guard, Rainsford..."
He had never slept in a better bed, Rainsford decided.
It is in the final two words that we finally discover the winner. The reader can assume that Zaroff either fell prey to the hounds or died at the hands of Rainsford since Rainsford would not have rested until the game was complete. Rainsford also proved to be the winner of the initial game when he managed to elude Zaroff by jumping from the cliff into the waters below. Rainsford won the first game according to the rules set down by Zaroff.
"I'll cheerfully acknowledge myself defeat if I do not find you by midnight of the third day," said General Zaroff.