Police say Winkle was spotted near Mill Avenue and Baseline Road on May 7 weaving in the roadway by an officer before being pulled over.
According to the police report, Winkle was also seen stopping abruptly at a red light. Winkle exhibited signs and symptoms consistent with impairment during the traffic stop, according to police.
Authorities said Winkle had a blood-alcohol level of 0.22 percent, nearly three times the state's legal limit for drivers.
Winkle previously pleaded guilty in Tempe Municipal Court and was sentenced to serve six days in jail and 49 days of home detention. He had a previous DUI conviction in Scottsdale in 2009.
A community economic development consultant and a co-founder of Mesa Urban Garden, Winkle was elected in 2016 and took office in January to represent a west Mesa district.
Winkle asked to be allowed to remain on the council. Winkle said that he attended Alcoholics Anonymous meetings and went to addiction counseling after his May arrest.
After the meeting, he said he was disappointed and saddened by the decision.
"But, it's not going to stop my work for the city," Winkle said. "I care about this city, I care what happens here. That's not going to change after this decision."
Mesa is Arizona's third most populous city, with an estimated 2016 population of 467,532, according to the Arizona Office of Economic Opportunity.
District 3 residents interested in serving their community can apply for a temporary appointment to the Mesa City Council through the city's website. The temporary Council seat term will end in the fall of 2018.
Applications will be accepted September 5 to 19. The council will vote on a candidate on September 18.