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The savings will come from the controversial £155m IT partnership with Capgemini finalised last month. A council spokesman said Capgemini would make the council more efficient and save money by reducing paperwork and duplication.
The £72m saved will be reinvested into front-line services such as education and the environment, the spokesman added.
The council plans to give schools access to its new IT systems to help improve their administration in areas such as finance, purchasing and human resources.
It also plans to build a social services call centre to take routine inquiries, such as providing information on the nearest care home or entitlement to benefits, the spokesman said.
Mary Jones, council cabinet member for top.performance and the service@swansea initiative, said, "Our plans have the potential to deliver savings and, at the same time, vastly improve council services.
"This means there will be less pressure on the council to raise council tax to pay for the improved services."