Anušauskas made the claim in an interview with the Baltic news site Delfi, according to the kresy24.pl website.
It quoted the Lithuanian defence minister as saying that the situation on the Lithuanian side of the border "has indeed become stable," with border officials as well as soldiers blocking migrants from crossing into the land.
"Our institutions are now handling those illegal immigrants who have found their way into Lithuania," Anušauskas said, as quoted by the Polish website.
"In other words, we have to make decisions about their further stay," Anušauskas elaborated, "and the migration service is drawing up documents, checking them, following all the procedures which apply in this kind of situation."
He argued this was "in stark contrast to the state of affairs on the Belarusian side," where he said he "wouldn't say things have stabilised," according to kresy24.pl.
"Probably around 4,000 to 5,000 illegal immigrants are still squatting in the forests of western Belarus," Anušauskas also told the Delfi news service, according to kresy24.pl.
He added: "Others are staying in hotels, but their money, as we know, is running out."
Belarusian law enforcement officials "are driving them to the borderland and forcing them to enter Lithuania," Anušauskas also said, stressing that "obviously we are not letting that come to pass," kresy24.pl reported.
The Lithuanian border service is coordinating with the military, Anušauskas told reporters, while a fence is being built along the boundary, at a rate of half a kilometre per day.
"In any case," the Lithuanian defence minister was quoted as saying, "we can see that Belarus' Lukashenko has himself plunged into the trap he had been preparing for the countries of the European Union."
A spokesman for the European Commission said on Thursday that the EU could move to impose new sanctions on Belarus amid a sharp increase in the number of migrants trying to enter Lithuania as well as Poland from that country.