More than 2,000 people were rescued on Friday and 3,000 on Saturday in dozens of separate rescues, the Italian Coast Guard said.
But at least seven people drowned as aid workers struggled to rescue more than 1,500 migrants in one operation.
An eight-year-old boy was among the dead, rescue workers said.
An earlier agency report said 20 bodies had been recovered by the Migrant Offshore Aid Station (Moas), but this was later corrected.
Moas said its rescue started in the early hours of Saturday and had continued non-stop into Sunday afternoon.