We would usually say "I won first prize". If you say "I won the first prize" it sounds like you won some prize that came up first, whatever that was. The term "first prize" indicates that you were the winner of a competition. The actual prize isn't really the focus.

Thank you, Nay.

Thank you, Elizabeth.

Either actually are ok honestly. If you are responding to a question, it would be more likely to be the first one I think, but they are both correct. You'd be more likely to say the 2nd though!

I was awarded first prize in the competition. Or I won first place in the competition .