Two cabinet ministers campaigning to leave the EU say the National Living Wage will attract workers from poorer EU states looking for a pay rise.
Chancellor George Osborne said it meant "Britain is getting a pay rise" and millions of workers would benefit.
But John Whittingdale and Chris Grayling said the best way to ensure British people were the main beneficiaries was by leaving the EU.
The £7.20-an-hour minimum wage will apply to workers aged 25 and over.
It is a 50p-an-hour increase on the National Minimum Wage of £6.70 an hour - to which workers aged 21 to 24 are still entitled.
The National Living Wage, announced in last year's Budget, is set to rise to £9 an hour by 2020, as part of government attempts to cut the numbers of low paid workers relying on in-work benefits, such as tax credits.
Mr Osborne said: "We said that Britain deserved a pay rise and today Britain is getting a pay rise."
David Cameron secured cuts to in-work benefits for new arrivals as part of his EU reform deal, claiming that it would reduce a major "pull factor" attracting migrant EU workers to the UK.
But exit campaigners have suggested that Mr Cameron's reforms will be rendered ineffective by the increasing level of the minimum wage which, they say, will be a pull factor in itself.
But Culture Secretary John Whittingdale, one of four cabinet members who want Britain to leave the EU, said the National Living Wage was a "great idea".
But he told The Times the policy "will fuel even higher levels of immigration" from EU workers seeking higher wages - putting a strain on public services.
Commons Leader Mr Grayling told BBC News that the National Living Wage was the "right thing for working people in this country".
But he added: "The Living Wage certainly makes it more attractive for somebody to move to the United Kingdom to get a job because it increases the differentials with other countries.
"The best way of addressing that is to ensure that we can put in place controls and of course those controls cannot be in place as long as we are members of the European Union."
The cabinet is split over the EU referendum with 17 full members in favour of staying in and four wanting to leave.
Asked whether the National Living Wage would increase EU immigration, Treasury Minister David Gauke said: "I don't think it will. I think you have to put it in the context of what else we are doing as a government, including the changes to in-work benefits."
And the Britain Stronger in Europe campaign accused Leave campaigners of an "inherent antipathy to using the clout of government to raise wages for the very lowest paid".
A spokesman added: "Leave campaigners have admitted that leaving the EU won't necessarily end freedom of movement. Norway and Switzerland, for example, are both outside of the EU but have free movement."
A government report on the National Minimum Wage, based on amounts at the end of 2015, suggested the UK's £6.70-an-hour for those aged 21 and older was among the highest in Europe.
But the government says it is not clear how other EU countries will increase their minimum wages and believes its changes to welfare rules will address the pull factor for EU migrants.