Just two weeks after going down 3-0 at home to Napoli, Claudio Ranieri's side lost again, despite taking a sixth-minute lead through Esteban Cambiasso.
Sergio Almiron and a Francesco Lodi penalty turned the game around in the second half, as the Sicilians took a deserved three points to leave the 2010 Italian champions with four defeats from their opening six games.
Last season's title winners AC Milan are not having the best of times either but they picked up only their second win of the campaign after downing Palermo 3-0 at home.
Antonio Nocerino broke the deadlock six minutes before half-time and Robinho (55) and Antonio Cassano (63) then added further goals to seal the victory.
The win lifts Milan onto eight points, four above Inter and three adrift of joint leaders Juventus and Udinese, who are both in action.
Saturday's other match saw third-placed Napoli miss the chance to take over at the summit as they lost 2-1 at home to Parma.
Massimo Gobbi put Parma ahead in the 57th minute and although Napoli drew level through substitute Giuseppe Mascara's 76th-minute goal, the visitors took the points when Francesco Modesto netted eight minutes from time.