The race was the wettest in the event's history, as more than 3 inches of rain fell Saturday.
Justify broke from the No. 7 post well, ran to the front with Promises Fulfilled and pulled away just as the horses passed the 3/4-mile marker in the 1-1/4 mile race.
Smith, who won the Derby on Giacomo in 2005, said the victory was due to the horse.
"My job was just to get him out of there," he said. "I just basically stayed out the way."