The new centre has 27 trained customer service agents to handle the calls.
The council hopes that by opening earlier and closing later, as well as introducing a new Saturday service, it will make it easier for people to get in touch.
The new centre uses Macfarlane CallPlus technology integrated with Lagan enterprise case management and Cisco Call Manager software to drive service improvements.
The centre handles council tax, housing benefit and care enquiries from the public.
The centre is the first phase of the council's Access to Services programme, a long-term initiative that aims to make it as easy as possible for customers to get in touch with the services they require.
Later phases of the project will see the opening of three face-to-face multi-services centres, one in each of the major towns in the borough, including Stockton, Billingham and Thornaby.
The council expects to answer about 180,000 incoming calls per year for the three services of council tax, housing benefit and care.