Plans include relocating the town's Central Library and cutting back on the Citizens Advice Bureau.
The council will also work with traders to find a private buyer for the town's Victorian indoor market.
The Labour-run authority said it was implementing the cuts "with a heavy heart" and blamed "severe austerity cuts imposed by central government".
Councillor Bill Dixon, leader of the authority, said that tough choices had been made, but local people had been able to comment during a public consultation.
"There were other cuts that people didn't want us to make," he said.
"Withdrawing school patrol crossings for example - some authorities have done that, this authority decided not to."
Responding to protesters opposed to the relocation of the Crown Street Library to the Dolphin Centre, he said:  "I was born and bred in the town.
"I do understand the attachment to Crown Street Library, but we should emphasise we are not closing the library but moving it."

The scheme, developed by the OU, has been designed to observe students' paths through courses and engagement with online learning modules.
The OU's Prof John Domingue told the BBC: "This has been developed as a tool for tutors to help students.
"We are planning to use this data for students studying in 2015-16."
The programme, called OU Analyse, was developed at the university during the 2013-14 academic year.
It uses a variety of data sets held by the OU and assesses the likelihood of a student submitting their next assignment by using information gathered from four different algorithms.
The more algorithms that indicate the student will not send in their work - the higher the chance of that happening.
Prof Domingue added: "We take advantage of the fact that modules are presented many times. One can use the experience of previous students to benefit future students. An interesting fact is that the data of the interactions before the course actually starts, like reading the material available and engaging with forums, is extremely valuable.
"Currently it is deployed on 13 modules, mostly level one modules in order to try to provide timely indicators that students may be struggling.
"The model is tailored to each of the modules and by using this and identifying students, more tailored support may be provided."
The OU has taken steps to ensure that students' privacy is not compromised by the scheme - putting in place an ethics policy, agreed in consultation with a student committee, to protect their data.
Ruth Tudor, president of the Open University's Students' Association, thinks the scheme is a positive one.
"It's a great idea and a great way of providing targeted support to students who may be struggling and need extra help," she told the BBC.
"I would like to think that this would improve the drop-out rate from OU courses. You must remember they take students who have come from no academic background so it is always possible that those people take on more than they think they can manage.
"That's why data analytics will help provide extra support and can help them be successful."
Reports suggest that some other British universities are thinking of using similar methods and Ms Tudor added: "I don't see why this shouldn't happen in other universities. It has happened in the USA and I'm surprised they're not using it here."

Described as "unusual debris" by Highways England, it was found on the southbound carriageway between junctions 30 and 29a in Derbyshire.
Fortunately there were no accidents and it was moved by 09:00 GMT having been there for 30 minutes.
It is thought to have fallen off a car roof and has since been reunited with its owner.
Sightings of the canoe provoked surprise on social media.
Anthony Temperton tweeted: "Anyone lost a canoe?! Last seen overtaking everyone on M1 j30 this morning."
Jeannette Morgan tweeted: "It's not every day you see a canoe in the middle of the M1!"
Glynn M tweeted: "It's not been wet enough to canoe on the fast lane of the M1."
Darlington Council has rubber stamped moves aimed at making £10m savings over the next four years.
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Students at the Open University are going to have their progress monitored by a set of algorithms to spot if they need any extra support.
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Part of the M1 was brought to a standstill as traffic officers removed a canoe from the outside lane.