Mr Mackay told MSPs at Holyrood he was prepared to engage with opposition parties on longer-term change as part of a “journey” on local taxation.
He was speaking in a debate on government proposals to bring in higher charges for some households in order to raise an extra £100 million a year for schools.
Under the reforms, set out by First Minister Nicola Sturgeon in March, the average band E household will pay about £2 more per week, with those in the highest band paying an extra £10 a week - an average of £517 a year.
The changes follow a report by the Commission on Local Tax Reform which last year called for an end to the council tax and urged politicians to implement a fairer, more progressive and transparent tax to fund local services.
Legislation to bring about the reforms has already been set out in parliament.
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Mr Mackay said: “I believe that these regulations will unlock finance for education as expressed in the SNP manifesto pledge, there will be protection in terms of the council tax reduction scheme changes as well.
“These initial reforms can be delivered at low administrative cost and achieve their purpose.
“Longer-term change, I think, will need more discussion, consensus and engagement, and I am certainly committed to that through the motion and through the engagement with political parties as we go forward.
“We have embarked on a journey in local taxation. We want to make it more progressive, deliver the steps that we got support for at the elections, then engage further on what can be delivered next in view of the report.”
Conservative Murdo Fraser said the SNP appeared to have “abandoned” a 2007 manifesto pledge to replace council tax with a local income tax.
Mr Fraser said the Tories supported an end to the council tax freeze, additional protections for low-income households and higher charges for those in properties in bands G and H, but opposed increases for homes in bands E and F.
He added: “Just as seriously, we oppose the approach that ministers are taking in relation to how the increase in council tax will be dealt with.
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“Ministers want to create a school attainment fund with money going direct to schools, that’s an ambition that we agree with.
“But they want to fund this by clawing back from councils that additional money - £100 million that will be raised by these council tax revenues and take this centrally to pay directly to schools.
“There is absolutely no precedent for what is currently being proposed which undermines both local democracy and local accountability.”
Responding to concerns about how education funds would be distributed, Mr Mackay said: “I can categorically assure every local authority area that every penny raised in council tax will stay in that local authority area.
“How we are proposing to allocate revenues towards education as we’ve proposed in the manifesto is through the revenue support grant.
“It is similar to business rates in terms of how that mechanism works and I don’t hear the complaint that that mechanism hasn’t worked to the satisfaction of local government.
“So, the principle is there but I’m very clear that which is raised at a local level through council tax will stay with those local authorities.”