Italian authorities allowed the boat to dock only after European countries agreed to take in most of the migrants.
Under an EU-brokered deal, 67 of them will go to four other countries, while the others will stay in Italy.
Germany and France will take 60 between them, Portugal will take five and Ireland will take two.
Italy will accept 21 of the migrants.
The boat is run by the German charity Sea-Eye.
A Taranto official, Gabriella Ficocelli, told the Italian news agency ANSA that among those migrants on board were five unaccompanied minors who were “tired and tried by the voyage”.
A doctor on board the ship said that were some people present who were suffering from old wounds inflicted by firearms and torture in the detention camps of Libya.
The migrants come from Nigeria, Gambia, Guinea, Cameroon and Senegal.
In an interview with RaiNews24, the Alan Kurdi shipmaster described the most difficult moment as “when we were rescuing the migrants at sea and the ship was suddenly surrounded by armed persons threatening us”.
This is the second disembarkation of migrants at the mainland port of Taranto, following the arrival of the Ocean Viking on October 16, which carried 176 migrants on board.