The champions, Milan, won 4-0 at home to Bologna to keep on their tails, alongside Fiorentina, who were held 1-1 at Udinese, Roma, who thrashed nine-man Perugia 3-1and promoted Reggina, who needed an 86th-minute goal to beat Piacenza.
After last week's 5-1 thrashing of Parma, Inter were quickly brought back to earth. Torino's Marco Ferrante squandered several first-half chances and saw Peruzzi save his second-half penalty before Vieri headed the winner from a free-kick by the Greek defender, Grigorio Georgatos.
Udinese's Paolo Poggi missed a sixth-minute penalty against Fiorentina and two minutes later the Argentinian, Gabriel Batistuta, finished off fine work by Pedrag Mijatovic for his fourth goal in two matches. The home side, however, avoided defeat with a 54th-minute goal from Stefano Fiore.
Roma's Fabio Capello stuck by his expensive summer signing, Vincenzo Montella, and the decision paid off as he scored in the 37th minute to launch the crushing of Perugia.
Reggina continued their remarkable first season in Serie A with a late winner to beat Piacenza, courtesy of a Bruno Cirillo header.
Fabian Valtolina put Venezia ahead at Cagliari after four minutes but the Belgian Luis Oliveira set up a 35th-minute equaliser for Daniele Berretta, earning Cagliari their first point of the season.Reuse content