HAVING SMOKED THE LAST CIGARETTE, the pack WAS THROWN AWAY by the man, and HE DECLARED THAT HE WOULD NEVER SMOKE AGAIN.
That is a a participial phrase. (A participle is an "-ing" word that describes another word.) When we have a participle phrase at the beginning of a sentence, our readers or listeners expect that phrase to describe the word that follows, usually the subject of the sentence. If the word which the phrase is intended to describe doesn't come next, we have a dangling participle, a modifying phrase that is disconnected from the word it is supposed to describe.
In the sentence you gave me, the key following word is "pack." It was not the pack that smoked the last cigarette but the man. If you think about it, you can see the pack somehow smoking a cigarette. Part of the problem with this sentence is that the verb is passive. An active verb would avoid the dangling participle. Having smoked the last cigarette, the man threw the pack away, etc.