Correct probably isn't a helpful word in this context. There are three possibilities: The doorbell rang. He had finished his meal (1); The doorbell rang; he had finished his meal (2): and The doorbell rang after he had finished his meal (3) *. All three are grammatical, and express the same sequence of events, but the implications are subtly different.
(1) implies no connection at all: The doorbell rang. The sun had gone down. At the Palace, the American President was making a speech. As such, it's probably bad style, unless you are simply setting the scene. (2) explicitly connects the ideas, and the reader would expect some explanation; perhaps he disliked interrupted meals, but was now prepared to answer the door. (3) emphasises the ring at the door, and puts it in a context.
That's the thing about English; there are usually several ways to put a sentence, but only one way to express your thought properly.
* Actually, there are many ways to rephrase it, but most of them boil down to one of the three.