Mr Mackay also confirmed he would not increase the basic rate of income tax in Scotland, even though ministers now have the power to do so.
He told MSPs at Holyrood: “In using the Scotland Act income tax powers for the very first time, we must have a balanced approach.
“Let me be clear, I will not pass the costs of UK austerity on to the household budgets of the lowest-income taxpayers.”
He did confirm the threshold for the 40p rate of tax would only rise by the rate of inflation in Scotland - meaning this higher rate will be paid by those earning £43,430 north of the border compared to the £45,000 in the rest of the UK.
Mr Mackay said that meant his budget would “not give a substantial real-terms tax cut to the top 10 per cent of income earners”.
He said he “sympathised” with calls for the 50p top rate of income tax to be reintroduced in Scotland for those earning £150,000 a year or more.
“I have had to balance that with the risk to our economy and am maintaining the current rate,” Mr Mackay told MSPs.
He insisted this approach had been “endorsed by the electorate” and was the “right thing to do for our economy, jobs and public services”.
The measures on income tax directly followed the proposals the SNP had already set out in the run-up to May’s Holyrood election.
However, Mr Mackay announced one change in tax, saying he had dropped plans to give the extra funds raised by upping council tax charges for those in the largest homes direct to schools in a bid to boost education.
This approach had been widely criticised by opposition parties at Holyrood and council leaders, who feared it would undermine local democracy.
Instead of the money coming from the revenues created by raising council-tax charges for properties in bands E to H, Mr Mackay said the Scottish Government would give £120 million to schools to be “spent at the discretion of teachers”.
“I will not sacrifice the educational chances of Scotland’s poorest pupils,” he said.
“I will not abandon our radical plan to give schools direct control over significant new resources.”
The ending of the council-tax freeze in April 2017 means from then local government leaders can up the charge by up to 3 per cent - potentially generating up to £70 million additional funds a year.
Mr Mackay told local authority leaders: “Councils will keep the full value of the revenue from council-tax rebanding - every penny raised locally will be spent locally as councils see fit.”
He said the Scottish Government would do this at the same time as delivering “our pledge to help schools close the attainment gape from central funds”.
Councils’ spending power will increased by £240.6 million, or 2.3 per cent, next year as a result of both cash from the Scottish Government and local taxation, the Finance Secretary said.
He told MSPs: “That is a settlement which invests in education, invests in social care and invests in local services.”
Overall, Mr Mackay said: “This is a budget for growth and public services, for our environment and our communities.
“It delivers increased investment in education, record investment in the NHS, protects low-income households from tax hikes and supports more and better jobs.
“It delivers £700 million of additional spending on our economy and public services. That is a budget for Scotland.”
The Finance Secretary said he was delivering his first Scottish budget in the midst of “challenging economic and political circumstances”.
He told MSPs that UK Government plans for a hard Brexit “represent a key risk to Scotland’s economy”.
Scotland’s GDP is forecast to grow by about 1 per cent in 2016-17 and 1.3 per cent in 2017-18, compared to about 2 per cent a year previously.
“These lower forecasts reflect the impact of the Brexit vote,” he said.