A total of 13 of the council's 40 seats were up for election.
Mr Slack said: "I am proud of what I have done for Exeter.
"If that is what the voters have decided I will have to take it on the chin."
Ms Newton said: "I think because the people of Exeter wanted change and I think because I've spent a lot of time the last few months meeting people and listening to people and getting things done.
"They saw that I was available and that I was going to be myself and that I have that level of honesty and integrity.
"It may be a small margin but an all important margin and I'm really looking forward to getting started."
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