Neither is incorrect, but the usual way to say this is "He has been elected chairman of the committee," as indicated by this Ngram.
I also would have tended to go with that way of saying it, but the Ngram helped me give evidence. I went ahead and checked with "president", "chairperson", "treasurer", etc, and came up with the same results each time. Just in case, I narrowed my search down to comparing "he was elected chairman" with "he was elected the chairman", and got the same result.
So much for popular usage. Let's now take a look at the meaning of "the" in that sentence.
Since "the" normally indicates either that the reader has had previous knowledge of the subject, or that the reader should know that it is the only one of its kind, saying "he was elected the chairman" makes sense. There is, after all, only one chairman, even if you've never heard of him before.
That said, the word "elected" does have its quirks. You might say that "elected chairman" and "elected president", though not at first obviously correct, has come to be a shortcut, since in most committees, clubs, etc, "the" is no longer necessary or helpful to get a meaning across.