Copenhagen forecasts that the deal could save at least 400 million kroner [$59.4 million], adding that the country could expect a further "tightening" later this year.
"For many this will make it impossible to be able to both pay rent in the cities and have money enough for food and clothes for themselves and their children," Kramp was quoted as saying by the Copenhagen Post.
Students in Denmark, many of whom are foreigners, are supported by the state, but immigrants would be left without similar possibilities, Kramp underlined.
In mid-June, general elections were held in Denmark, with the issue of immigration high on the agenda.
The government, led by the center-right Venstre party is supported by the far-right anti-immigration Danish People’s Party (DPP). During the election campaign, Venstre blamed budget cuts, unemployment and social problems on immigration, promising to "take action" on refugees.