Zoro first fights Mihawk in the East Blue region of One Piece, Episode 24 with the title “Hawk Eyes Mihawk! Swordsman Zoro Falls into the Sea!”
WATCH THIS EPISODE
In this episode Nami Steals the “Going Merry”, Luffy who wants Nami as his navigator asks Zoro to go and get her. Just at that moment Zoro catches glimpse of a small coffin like boat not too far form him. Seeing that it was Dracule Mihawk the greatest swordsman in the world aboard it, Zoro recklessly challenges Mihawk in order to fulfill his promise to Kuina. Mihawk replies that he is not an idiot who would use all his strength to hunt a mere rabbit. Mihawk then pulls a small pocket knife from around his neck which was hanging as a pendant to challenge Zoro. Zoro was surprised to find out how much stronger than him Dracule Mihawk was, so much so that he questioned if there was really that much gap between the both of them. Mihawk easily fended off Zoro’s attacks with his pocket knife and even got to injure Zoro with it. Surprised by his will, Mihawk takes out his Yoru. Even Zoro’s strongest attacks were treated like child’s play, and he eventually lost two of his swords to breakage. Knowing he had lost, he saw it as on honor to die by the sword and so he was slashed with the Yoru.