The biggest ever list of offenders included employers in hairdressing, retail, hospitality and care homes, as well as the retail giant.
Excuses for underpaying workers included using tips to top up pay, docking wages to pay for Christmas parties, and making staff pay for their own uniforms.
Debenhams was accused of failing to pay almost £135,000 to just under 12,000 workers.
The company said it made a technical error in its payroll calculations, which resulted in an average underpayment of around £10 per person to affected workers in 2015.
A spokesman said: “As a responsible employer Debenhams is committed to the national minimum wage, and as soon as the error was identified by a routine HMRC audit last year we reimbursed all those affected.
"We have apologised to all our colleagues affected and have taken steps to ensure it cannot happen again.”
The Business Department said more than 1,500 cases are being worked on by HM Revenue and Customs, with more firms set to be named.
The new announcement means that more than 1,000 employers have been identified since the policy started in 2013.
The new list includes 84 employers in the hospitality industry, 51 in retail, 39 hairdressers and 24 in social care.
Business minister Margot James said: "Every worker in the UK is entitled to at least the national minimum or living wage and this government will ensure they get it.
"That is why we have named and shamed more than 350 employers who failed to pay the legal minimum, sending the clear message to employers that minimum wage abuses will not go unpunished."
Unions welcomed the announcement, but called for more prosecutions - there have been 13 since 2007.
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TUC general secretary Frances O'Grady said: "This should be a wake-up call for employers who value their reputation. If you cheat your staff out of the minimum wage you will be named and shamed.
"But we also need to see prosecutions and higher fines for the most serious offenders, especially those who deliberately flout the law."
Unison general secretary Dave Prentis said: "This list fails to shame the larger care firms who are equally guilty of denying staff a fair wage. Those in the spotlight today are just the tip of the iceberg.
"HM Revenue & Customs needs to get much tougher with more inspections to identify scrooge employers."
Unite assistant general secretary Steve Turner said: "The Government needs to crack down further on employers who failed to pay the national minimum wage to some of the most low-paid and vulnerable workers in the country.
"The fact that the Government has mounted only 13 prosecutions for non-compliance since 2007 is pathetic.
"In America, bad bosses are jailed and heavily fined for 'wage theft' which is what this is, exploiting workers in such a shameful fashion."
Tony Fitzpatrick, chief executive of St Mirren Football Club, which is named for failing to pay £1,277 to one worker, said: "I don't know why we are on the list. All of this was dealt with some time ago.
"It must have been before we came in or before the take-over."
Shadow business secretary Rebecca Long-Bailey said: "It is frankly disgraceful that a record number of employers have failed to pay their workers the basic minimum wage, and hopefully the repercussions of being publicly named and shamed will act as a deterrent for other unscrupulous employers.
"However, the fact remains that the current so-called national living wage is that in name only.
"The Chancellor even announced a cut in the rate at the Autumn Statement last year, leaving 2.7 million people over £1,300 worse off by 2020.
"Labour will deliver a real living wage so that every worker earns enough to really live."
Iain Wright, chairman of the Business, Energy and Industrial Strategy Committee, said: "If companies are failing to pay the National Minimum Wage, leaving low-paid workers short, then it's right that those companies are named and shamed.
"But the problem of non-payment of the NMW surely goes far beyond the 350 companies named today.
"Evidence to our current world of work inquiry suggests this latest HMRC action is just the tip of the iceberg.
"We need tougher enforcement of the NMW, the question is do the HMRC and the Low Pay Commission have the powers and resources to do the job?"