Johnson will have the charges dropped if he pays court costs and completes 363 days of unsupervised probation after agreeing to a judicial diversion program.
He remains subject to NFL discipline.
Johnson was charged with a DUI and cited for simple possession in July 2017 after police stopped him for a traffic violation and found he had bloodshot eyes with alcohol on his breath and a marijuana cigarette next to him.
Jurors in Knoxville, Tennessee, acquitted Johnson and former University of Tennessee teammate Michael Williams of aggravated rape charges 2 1/2 months ago, stemming from a November 2014 incident.
The Broncos signed Johnson on Aug. 13, following his trial.