The external structure of this story, with its division into three parts, is complicated. Because the events do no unfold in chronological order, the exposition-through-denouement pattern is also not chronological. To trace the dramatic structure, it becomes necessary to arrange events in chronological order. Here's the chronology of events in actual order:
Peyton is described as a loyal Southerner who longs to participate in the war. He speaks to a soldier whom he assumes is a Confederate. The soldier is instead a Union scout who gives Peyton the idea of burning Owl Creek Bridge. Peyton attempts to burn the bridge and is captured. As the Union soldiers prepare to hang Peyton, his fear affects his mind as he surveys his surroundings. He is hanged. As his body falls into Owl Creek, his mind acts to protect him at the moment of his death as he fantasizes about escaping and returning home to his wife's arms. His falling body hits the end of the rope, breaking his neck. His dead body then swings back and forth from the end of the rope below the bridge.
In this chronology, the establishment of Peyton's character would serve as exposition. His discussion with the Union scout would comprise rising action which would continue until the moment Peyton reaches out for his wife's arms. The sharp pain he feels in his neck represents the turning point or dramatic climax of the story:
As he is about to clasp her he feels a stunning blow upon the back of the neck . . . .
The falling action makes the reality of Peyton's situation clear:
. . . a blinding white light blazes all about him with a sound like the shock of a cannon--then all is darkness and silence!
Finally, the last paragraph of the story serves as its denouement:
Peyton Farquhar was dead; his body, with a broken neck, swung gently from side to side beneath the timbers of the Owl Creek bridge.
By structuring his story in such an unusual way, Bierce creates a very shocking--and ironic--conclusion.