tl;dr: Utena probably wins because of some sort of spiritual or moral quality she has.
Why is Utena able to win?
In Utena's second duel, Utena is losing until she "absorbs" power from the prince. Moreover, Touga, who is watching the duel, says:
What was that just now?! Was that the power to bring the world revolution? The Power of Dios?!
This indicates that Utena is probably aided by something that has nothing to do with her sword skills---probably whatever the image of the prince signifies, as the image frequently appears falling from the sky or imposed on Utena's body just before she deals a finishing blow.
Incidentally, in ep. 15, Mamiya says, of the Student Council members:
Swords strong enough to defeat Utena Tenjou may have crystallized inside of their hearts!
Mikage counters this by saying that the "weak-hearted fools" on the Student Council have already been defeated by Utena before. The emphasis here isn't on the physical abilities of the duelists. Thus, there might be some "spiritual" or "moral" aspect to some of the duels.
I should also note that in some of the cases, it's not surprising that Utena is able to beat the duelists, even if we don't consider any "unnatural" causes. For example, in ep. 5, Utena wins the duel after Miki becomes distracted (when Anthy cheers on Utena). Similarly, in ep. 29, Juri leaves the duel after her locket is broken.
Who is the prince?
In Saionji and Touga's ep. 33 conversation, Touga says that Akio saved Utena as a child by showing her "something eternal".
Later, in ep. 34, the flashback to Utena's childhood in ep. 34 indicates that the prince who saves Utena is the prince whose image appears in Utena's duels. This is the Rose Prince, who in second flashback is revealed to be Dios and Anthy's brother.
Akio must then be a degraded version of the prince, or a "fallen" Dios---in the flashback, it is stated:
the prince [the Rose Bride] loves is no longer the prince she knew. Not anymore. He is now the End of the World.
Thus, Touga might be right about Utena winning with the "Power of Dios". (Being not entirely sure what the "Power of Dios" is, I hesitate to say that this completely explains Utena's victories.) If so, perhaps she is channeling this power towards the end of the duels, when the image appears.