Scotland's new system of income tax rates and bands has come into force.
Scottish income tax - how changes affect you Budget 2018: income tax calculator
He said: "This progressive approach to reforming income tax will not only protect the lowest earning taxpayers, but ensure 70% of Scottish taxpayers pay less tax this year than last year for a given income, while the majority of Scottish taxpayers will pay less than if they lived elsewhere in the UK.
"These measures, combined with our investment in the NHS, the economy, infrastructure, education and essential public services, ensure Scotland will be the fairest taxed part of the UK and provide the best deal for taxpayers."
'Unimaginative and unnecessary'
The budget ultimately won the support of the Greens and two of Holyrood's five Lib Dem MSPs, but the Conservatives and Labour both voted against the overall package - for very different reasons.
Tory finance spokesman Murdo Fraser said the changes were a "punishing move which businesses and experts right across the country warned against".
He said: "Scotland's economy is already suffering at the hands of SNP stewardship, growing at less than half the rate of the UK average.
"The SNP's unimaginative and unnecessary tax grab will only make that worse. Instead, we should be striving to create a high wage, low tax, low welfare economy."
Labour set out its own, more radical tax proposals shortly before the first budget vote, and MSP James Kelly said the government's reforms were "nothing but a missed opportunity".
He said: "Derek Mackay has done nothing but tinker around the edges, passing Tory austerity on to the people of Scotland.
"Rather than asking the richest to pay their fair share, the SNP has cut council budgets and failed to deliver a proper pay rise for public sector workers. Scottish Labour would use the tax powers to make the richest pay their fair share, end austerity and deliver real change."
The Greens agreed a deal with Mr Mackay to back his budget along with the tax proposals, and persuaded him to reduce the threshold for the 41p band.
Co-convener Patrick Harvie said it was his party's approach which had "shifted the debate" on taxation in Scotland, declaring himself "delighted" that Scotland had a "radical opportunity to build a fairer tax system" with "progressive values".
The Lib Dems also called the budget a "missed opportunity", with leader Willie Rennie saying it "fails to invest the necessary funds to transform education and make a step change in mental health services".
However the Lib Dem MSPs for Orkney and Shetland - Liam McArthur and Tavish Scott - ultimately voted for it in order to back funding for northern isles ferry services.