The member for Hollinwood replaces Jim McMahon who won the Oldham West and Royton by-election last month.
Cllr Stretton, who has served as deputy leader since 2014, was chosen over Cllr Arooj Shah, who represents St Mary's ward.
She was elected in 2010 and is chair of the health and well being committee.

The True Blood star died after years of suffering from drug and alcohol abuse.
In a statement, his family said Ellis, 39, was ashamed of his addiction and did not want to talk about it.
But they added: "His family, however, believes that in death he would want his life to serve as a cautionary tale in an attempt to help others."
Speaking to the Hollywood Reporter through his manager Emily Gerson Saines, the family's statement continued: "Nelsan was a gentle, generous and kind soul.
"He was a father, a son, a grandson, a brother, a nephew, and a great friend to those that were lucky enough to know him."
Talking about the circumstances behind Ellis's death from heart failure, the statement explained: "After many stints in rehab, Nelsan attempted to withdraw from alcohol on his own.
"According to his father, during his withdrawal from alcohol he had a blood infection, his kidneys shut down, his liver was swollen, his blood pressure plummeted, and his dear sweet heart raced out of control."
Ellis played the flamboyant Lafayette Reynolds, a charismatic gay medium who was able to contact ghosts, in the HBO horror-drama series.
He appeared regularly throughout the series of True Blood after first appearing as the cook at a local restaurant in 2008.
He also featured alongside Viola Davis and Octavia Spencer in the film adaptation of Kathryn Stockett's novel The Help in 2011.
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Oldham Council will be led by a woman for the first time in its history, after Jean Stretton was selected by Labour councillors.
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Actor Nelsan Ellis's family has spoken about the actor's death on Saturday, in an effort to help other people who suffer from addiction.