Higgins triumphs despite an entirely selfish attitude, one that is bred in his secure position in life. He has all that he needs or seeks; when a new challenge erupts, he chooses to bring this new challenge into his world without at all modifying his world to meet the challenge. This almost results in failure at the ball, but in the end, success results because of Eliza Doolittle's sheer energy and willpower; and perhaps even because of her love for Higgins.
Eliza Doolittle changes into a Queen of Sheba, yes, but not through Henry Higgins' doing. His heavy-handed pedagogical techniques send her yearning for her simpler days in Covent Garden. In fact, there is no concrete proof in the play that his techniques motivate Eliza at all. She is not propelled to excel because of any of Higgins' outbursts or cruelties.
Eliza and Higgins are operating on their own wavelengths, and it is the sheer willpower of Eliza that actually functions to turn her into a Queen of Sheba. That is why Higgins finally falls for a younger woman in Eliza. As he tells his mother early on when asked to fall for a younger woman, Oh, I can't be bothered with young women. My idea of a loveable woman is something as like you as possible. I shall never get into the way of seriously liking young women: some habits lie too deep to be changed. [Rising abruptly and walking about, jingling his money and his keys in his trouser pockets] Besides, they're all idiots." (Act III)
Eliza succeeds despite Higgins, or in spite of Higgins, and Higgins recognizes this at the end. That is why he falls for her; here is a woman and a situation that he could not handle, and this is the sense of urgency for which he sought.