MARGAH, Afghanistan — The gunshot sounded like splitting wood. The American paratroopers, scattered across the hills of this rough border town in eastern Paktika province, dove for cover behind rocks, mud walls and irrigation ditches. A few seconds later, they trained their weapons on their attackers’ location.

The ensuing firefight — pitting the soldiers of Fox Company, 2nd Battalion, 506th Parachute Infantry Regiment plus their attached Afghan troops against a small band of Taliban fighters — was the climax to a suspense story told in reverse on April 8. The Americans’ 13-hour patrol began wi...