Joan Sturgeon’s decision to step down comes after the First Minister’s father, Robin, failed to be elected as a councillor last week.
Labour’s Louise McPhater won the Irvine West by-election on Thursday sparked by SNP councillor Ruth Maguire’s election as an MSP.
The win gave the Labour group more elected members than the SNP by 12 to 11. The remaining seven councillors are six independents and one Conservative.
In a statement, the SNP group said: “Given that the SNP group no longer command the largest representation on the council, it is only right that Labour be given the opportunity to form the next administration of the council.
“Accordingly, the SNP group has taken the difficult decision to resign from administration to allow Labour the opportunity to form an administration.”