While "No Such Thing" by Hal Hartley is not, by far, my favourite film, it's one I'll never forget. I was captured right from the opening monologue which began with, "I'm not the monster I used to be."
Not the greatest monologue in film history, nor the greatest character, but one of the great introductions to a character and an impactful way to introduce that character. If I had seen only this introduction, I would definitely have wanted to see the film after. Lucky for me, I saw it all at once.
While it is not incumbent on a film to be as stark or shocking off the top to attract me, here's a distinct case where an opening monologue, and perhaps an opening line alone, captured me.