The sea connected to the Mediterranean by the Dardanelles, Sea of Marmara, and Bosporus is the Black Sea.
This waterway is a chain of connected seas and straits: the Black Sea leads through the Bosporus into the Sea of Marmara, then through the Dardanelles into the Aegean Sea and the wider Mediterranean. The entire route lies in and around Turkey, with the Bosporus separating the European and Asian sides of Istanbul.
The passage has been strategically important since antiquity because it provides the Black Sea's maritime route to the Mediterranean and beyond. Modern commercial shipping uses it to connect ports in Turkey, Ukraine, Russia, Georgia, Romania, and Bulgaria with global trade routes.
The Aegean Sea is the first major sea south of the Dardanelles, so it can be mistaken for the answer. The Adriatic lies farther west along the Mediterranean, while the Caspian Sea is an enclosed body of water with no natural connection to the world ocean.