"The fact I won in every category allowed me to _______ him." 
(Because I won, I took the position of being above him.)"

This example made me think of the verb form of best:

: to get the better of : outdo

The fact I won in every category allowed me to _______ him.
(Because I won, I took the position of being above him.)

That example made me think of the verb best, meaning "to get the better of; outdo."