Question

Who won the Peloponnesian War?

Answer 1

Our two option are Athens and Sparta, as you know. Read this paragraph about the final result of the war:

However, the democratic leaders refused Spartan peace offerings, and the war continued at sea with the Spartan and Athenian fleets trading costly victories. The end came in 405 when the Athenian navy was destroyed at Aegospotami by the Spartan fleet under Lysander, who had received much aid from the Persians. The next year, starved by an impenetrable blockade, Athens capitulated. Athens’ defeat was perhaps the worst casualty in a war that crippled Greek military strength, and thus the most culturally advanced Greek state was brought into final eclipse.

"Athens capitulated. Athens' defeat was perhaps the worst casuality..."
Notice phrase "Athens defeat." Since Athens lost, the only other option is Sparta.

Sparta won the Peloponnesian War.