Inter captain Martinez has now scored nine goals in seven Serie A matches and holds a clear lead as the league's top scorer.
While Inter dominated the match's early stages, Salernitana gradually found their footing and managed to control the tempo in a goalless first half.
Martinez opened the scoring for Inter in the 62nd minute when he expertly collected Marcus Thuram's low cross into the box and delicately chipped the ball over Salernitana keeper Guillermo Ochoa.
Martinez completed his hat-trick from the penalty spot with five minutes remaining, after a foul by defender Matteo Lovato, who had pulled Thuram's shirt.
Inter host a Champions League group-stage match against Benfica on Tuesday, followed by a Serie A home game against Bologna next Saturday.
Persons:
Stadio, Inter Milan's Lautaro Martinez, Alberto Lingria, Lautaro Martinez, Martinez, Marcus Thuram's, Guillermo Ochoa, Mateusz Legowski's, Nicolo Barella, Matteo Lovato, Carlos Augusto's, Simone Inzaghi, Tommy Lund, Toby Davis, Clare Fallon
Organizations:
Soccer, Salernitana, Inter Milan, Stadio Arechi, Inter Milan's, Rights, Serie, Inter, Milan, Lazio, Sky Sport Italia, Champions League, Benfica, Bologna, Thomson
Locations:
Salerno, Italy, San Siro, Gdansk