This phrase doesn't really mean "I don't want any lecturing," because that sounds kind of negative. "You had me at hello," is always positive - it means that I agreed with you (or even fell in love with you!) at the very first instant you spoke, and nothing you said afterwards can make me feel any more positive towards you or whatever idea you're promoting. So while this phrase doesn't necessarily mean "charm" in the romantic sense, it's definitely more positive than just "to persuade," which is why @user45266 used "you won me over" as a synonym.
The reason why at is part of the catchphrase is because what you are emphasizing is the position of the words that won you over (i.e., the very first one or ones), so you use at to mean "expressing a particular location or time." Using with is grammatical, but does subtly change the emphasis to stressing which word it was rather than stressing the fact that it was the opening words.
If you want some equivalent expression that's less positive, you could say, "I got it already," which says that you already understand something and can be used in an exasperated way to mean, "I don't want any more lecturing."