They are the canon couple/fated lovers of Evangelion. They literally get together in every iteration of Evangelion.
NGE makes the romantic relationship apparent from Asuka's first episode - Shinji peeks at Asuka while changing (he would never do that to Rei), the flirting about being inappropriate, Asuka's hands on top of Shinji's, and their sync ratio. The entire series is intertwined with the webbing of their relationship - Shinji's admiration for Asuka's reason to pilot Eva, Asuka's jealousy when being replaced by Rei, Shinji selfless act of bravery to Asuka, the Kiss, Asuka's waiting for Shinji at the hospital, or the mind-rape scenes.
Compare that to the other characters in the show. Looking for a relationship there is someone who can't see the forest from the trees. Rei is motherly/platonic love. Misato is a substitute mother (who has her own problems and is rejected purposely by Shinji). Hikari belongs to Toji. Kaworu is an angel who can't understand humans. There might be an occasional snippet of the relationship between other actors, but you have to basically ignore the entire series to do so.
So why do they work? Its simple. The harsher the tsun, the sweeter the dere. And when Asuka goes sweet on Shinji, he can't resist.