The government will start actively naming and shaming employers who flout national minimum wage laws this autumn, the Department for Business, Innovation and Skills (BIS) has announced.

The policy could embarrass hundreds of companies which pay below the minimum wage of £6.19 for people over 21.

In the last tax year HM Revenue & Customs, which enforces minimum wage laws, found 736 employers who were paying staff too little, leading to the recovery of £3.9m in unpaid wages for more than 26,500 workers.

The recovery process, however, almost always remains secret with HMRC unable to confirm that inspectors have targeted large multinationals and high street chains even when they issue rebates to former staff.

Since 2011, when rules were first relaxed to allow BIS officials to name and shame the worst offenders, just one company – a hair and beauty salon in Leicester – has been exposed.

Unions and campaigners hailed Friday's announcement as a step forward but called for heavier court action.

The Guardian has learned that new employer codes for internships due to be issued by BIS in October are likely to recommend that interns are paid a proper wage.

On the Today programme, the business minister Jo Swinson admitted that the current system of issuing financial penalties had not sufficiently deterred companies from denying workers a legal wage. "Certainly the vast majority of those if not all should have been named," she said.

"I am introducing these changes to make sure that we do pursue employers that aren't properly living up to the law and paying their workers a proper wage. And of course it's quite right that when you have a system, that you review and see if it's working, and if it's not working then you make changes and work out what you can do to improve things, and that's exactly what we're doing."

Swinson said it was unfair to businesses that respected the law not to penalise those that flouted it. "I understand that employers won't want to be named, but of course we're only talking about employers that are breaking the law and the vast majority of employers are law-abiding, pay their employees properly, and therefore they shouldn't be disadvantaged by some of their competitors that might be getting away with paying less than the minimum wage."

Frances O'Grady, the TUC general secretary, said: "It is right to name and shame the minimum wage rogues, so that other employers who think they can get away with paying illegal poverty wages get the message loud and clear that cheating does not pay. At the moment all employers who have been found guilty of cheating workers out of a legal wage have to pay a financial penalty, but as this takes place behind closed doors justice is not seen to be done.

"But naming and shaming won't be enough to deter those employers who think they are above the law. Only a handful of employers have been taken to court since the minimum wage was introduced in 1999, yet over the years thousands of workers have complained to the minimum wage helpline that they are being ripped off.

"Employers need to know that there will be no hiding place if they break the law. The government must put more money into enforcement so that there are fewer places for even the most determined minimum wage cheats to hide."

Tanya De Grunwald, a campaigner with Graduate Fog, welcomed the move. "For some time HMRC has had a misplaced loyalty towards employers exploiting their workers. The priority of the whole system seems to be in keeping these employers details anonymous rather than in protecting the workers themselves. So hopefully this will scare employers who might be thinking of taking advantage of vulnerable young workers to abide by the law," she said.

Gus Baker of Intern Aware said it was time that offending companies were hauled into court as routine. He said: "It's fantastic that Swinson has agreed to name and shame employers who break the minimum wage. However, small fines and public embarrassment aren't enough to discourage the exploitation that is occurring. The government needs to properly get its act together and start prosecuting the companies that are breaking the law and not just telling them off."