Let's examine the outcome of the challenge between Sir Gawain and the Green Knight.

In one sense, the Green Knight absolutely is the winner. Not only does he withstand the blow from Gawain, surviving decapitation, Gawain does come to the Green Chapel at the prescribed time to submit to a similar blow, when it would have been easy for the Green Knight to kill him. So, if "winning" is associated with the recognition that the Green Knight is more powerful than Gawain, then the Green Knight wins.

This view doesn't change much when we learn that the Green Knight has been testing Gawain's virtue in addition to his courage. The Green Knight is still in a position of power, and Gawain still submits.

Ultimately, however, the Green Knight's challenge serves to confirm the moral superiority of Gawain. Even though he is less powerful physically, Gawain's self-reproach and relinquishing of the green girdle is an instance of moral power. His ability to stick to his knightly virtues even in the face of certain death is a testament to the achievement of Arthur's court. So, in this sense, Gawain is the winner: even though he lacks the Green Knight's supernatural power, he escapes with both his life and honor intact.