Annual increases will not rise above 2%, in line with a decision by Chancellor George Osborne for business in England and Wales.
Mr Swinney also expanded the Small Business Bonus Scheme.
The threshold for companies with multiple properties is being increased from a rateable value of £25,000 to £35,000, bringing an extra 4,000 additional properties into the discount scheme.
Mr Swinney, setting out the plans in the Scottish Parliament, said: "Business rates play a part in attracting and retaining businesses in Scotland.
"In the autumn budget statement last week, the Chancellor announced that the annual increase in the business rates poundage in England and Wales would be capped at 2%.
"To maintain the competitive advantage enjoyed by Scottish businesses and maintain Scotland's position as the best place to do business, we will match that cap and restrict the increase in the rates business pay to 2%.
"And to reinforce our commitments detailed in Scotland's Future, we will go further in our support for small business.
"Our Small Business Bonus scheme already supports over 92,000 premises. We will expand the Small Business Bonus Scheme for the rest of this parliament to help even more small businesses."
Andy Willox, of the Federation of Small Businesses (FSB), said: "The small business bonus continues to give enterprises in Scotland a real advantage.
"Over the last few years, our members have faced spiralling overheads and tough credit conditions. This scheme has injected Scottish smaller enterprises with working capital at a time when they needed it most. Firms that operate from multiple small premises, or are looking to expand into their second premises, alongside the FSB, will warmly welcome this announcement."
The 2% cap gives firms in Scotland a "level playing field", he added.
Laura McMahon, of the Confederation of British Industry, welcomed the changes for small firms, but said: "It is disappointing that the return to a level playing field on rates for larger retailers remains elusive. Larger retailers are still burdened with a Scotland-only rates surcharge of £95 million, which simply doesn't apply down south.
"Our retail members tell us that this levy is making them think twice about expanding in Scotland, particularly when they have options to invest elsewhere. After all, why build a new store in Dundee, if you can build one in Doncaster where the rates regime is more accommodating?"
Liz Cameron, chief executive of Scottish Chambers of Commerce, said the cap will save businesses millions of pounds next year.
"This is a welcome mitigation of the original plan to raise business rates by 3.2% next year," she said.
"Equally welcome is the extension of the threshold of the Small Business Bonus Scheme for businesses with multiple properties, which will assist an additional 4,000 businesses across Scotland."
But concerns remain about the "burden" of rates.
"Many businesses are concerned that there is a deep lack of fairness and transparency at the heart of the system of business rates, not just in Scotland but throughout the UK," she said.
"The Scottish Government consulted on the rating system earlier this year and some positives have come out of that review but more fundamental reform may now be required to ensure that business rates are fair and seen to be so by businesses."
Meanwhile, the Scottish Government has been urged to follow Westminster's lead after UK ministers announced a £1,000 rates discount for small retailers.
The reduction, unveiled as part of Mr Osborne's autumn statement, will apply to retail premises south of the border - including pubs, cafes and restaurants - with a rateable value of up to £50,000.
Tory finance spokesman Gavin Brown has now called on the Scottish Government to follow suit and introduce a similar scheme to help struggling retailers here.
He said the business rates reduction for retailers was one of the "big announcements" in Mr Osborne's mini-budget and would be worth about £350 million to the sector.
"This is a big shot in the arm for retail south of the border," the Tory said.
"It has been widely welcomed by businesses across the land, and our challenge to the Scottish Government is what do they intend to do to help retail in Scotland?"
The Conservative added: "We see them incessantly complaining about the powers they do not have while refusing the powers that they do have.
"Let's have some more action from the Scottish Government, let's use the powers they do have."
Mr Brown said the SNP Government's existing small business bonus scheme - which reduces rates for small firms - was an "excellent policy", but he added that only companies with a rateable value of £18,000 or less could take advantage of this, while the UK Government initiative would help those with a rateable value of up to £50,000.
Mr Brown went on: "The challenge to the Scottish Government has to be what are they going to do in response to that? What will retailers in Scotland get to match that, because retailers in Scotland I think have had a particularly difficult time as a sector."
The Tory said that when the SNP came to power as a minority government in 2007, it had helped companies with the introduction of the small business bonus scheme and its town centre regeneration fund.
But he added since the SNP had formed a majority government in 2011, "things have moved backwards" with a levy introduced on large stores selling alcohol and tobacco, and "huge increases to taxes on empty properties".