Unless you are going to exclude The Last of the Mohicans, The Scarlet Letter, and Moby Dick from the category of American novels, it is not possible to argue that The Adventures of Huckleberry Finn is chronologically the first. Indeed, it is not even Mark Twain's first novel: he had been writing them for more than a decade and published three before Huckleberry Finn appeared in 1884. However, although it was not the first novel he published, Huckleberry Finn is widely regarded as Twain's masterpiece, which is to say first in terms of quality. The word "first" is frequently used in this sense, to refer to precedence: clearly the First Lady is not the oldest lady in America.
The first American novel in this sense is close, if not identical, to the frequently discussed category of the Great American Novel. If we think of this as something like a foundational national epic, equivalent to the Aeneid in Roman literature, which defines America, then The Adventures of Huckleberry Finn has an excellent claim. Its nearest rivals are probably Moby Dick and The Great Gatsby. However, Moby Dick is primarily a novel of the sea, most of which does not even take place in America, while The Great Gatsby deals primarily with a small and rarefied section of American society living on Long Island. As a great, sprawling national epic celebrating the spirit of America and traveling across it, Huckleberry Finn has no rival except Leaves of Grass, which is not a novel, though it might be regarded as America's first national poem.