The 19-year-old Logano started on the pole, but fell well back after an incident with Greg Biffle early in the race. Logano steadily worked his way back to front and caught a break when Biffle and Denny Hamlin collided with less than 10 laps to go.
Logano took the lead following a caution with two laps to go, then pulled away from Vickers.
Carl Edwards finished third and pulled within 155 points of series leader Kyle Busch, who exited his No. 18 Toyota early in the race because of a fever. Hamlin replaced Busch and battled for the lead before getting caught between Biffle and Brad Keselowski, relegating the car to a 31st-place finish.