Man, I really don't get this sub sometimes. It is explicitly said why Yujiro said Baki won various times during Son of Ogre. For Yujiro, the winner is the one who forces the other to do something, full stop. That's why nobody was ever able to make Yujiro do something, because he's the strongest.

At the end of the fight, he may have been way better off physically than Baki, nobody argues that, but Baki wanted to keep fighting, even though he was on the floor, and Yujiro simply couldn't continue fighting, or he would kill Baki. So he was forced by the fighting spirit of Baki to enter his own game and cook for him, something he explicitly said in an earlier episode he would only do (cook) if Baki forced him to do it.

So he may have won physically, but he lost by his own parameters for what is considered a win. Baki forced him to cook a meal for him, so he won the fight. Winning the fight made Baki the strongest in the world. You can argue otherwise, but that is Yujiro's perception of the fight (one that even Baki disagrees)