A statement from Prime Minister Benjamin Netanyahu said that two women and a child were to be allowed in and that the rest of the group they were with, about 17 other African migrants, were to return to where they came from.
A government official who asked not to be named said the three were let in on humanitarian grounds. Israeli media said one of the women had been pregnant and had lost her baby.
About 60,000 African migrants have entered unlawfully in the past few years through the porous Egyptian border and have alarmed Israel, which says they are mostly job-seekers who threaten the demographics in the Jewish state of 7.8 million.
To curb the influx, Israel has been building a fence along the 260 km-long (160 mile) frontier with Egypt's Sinai desert and it is due to be completed by the end of 2012.
"We are determined to stop the flow of the infiltrators," Netanyahu said. "We have built a fence to achieve that goal ... and we will continue our efforts to return the infiltrators to their countries of origin."
Humanitarian organisations in Israel say the migrants should be considered for asylum and some Israelis have been troubled that their country, founded by war refugees and immigrants, should be packing off foreigners en masse.
Israeli media said the group of 20 migrants were Eritrean and that they arrived at the guarded border last Thursday. Israel's military said it had provided them with food and water. (Writing by Maayan Lubell; Editing by Louise Ireland)