Champions Juventus beat Lazio 2-0 and last season's runners-up Napoli came from two goals down to rally past AC Milan 3-2 on Saturday to both start their campaigns with the maximum six points.
Inter Milan — who returned to the Champions League after finishing fourth in Serie A last season — had fallen to a shock 1-0 opening defeat to Sassuolo.
The northerners looked in control in the San Siro with their Argentine top scorer last season Mauro Icardi setting up Croatian World Cup finalist Ivan Perisic for the opener after six minutes with Stefan de Vrij heading the second after 32 minutes.
Higuain joins Milan as Bonucci returns to Juventus
But Spalletti's side lost their way after an hour with Andrea Belotti pulling one back for Torino after a goalkeeping error by Samir Handanovic. New French signing Soualiho Metie earned the point on 68 minutes.
Salvatore Sirigu denied Perisic late on to give Torino their first point after last week's 1-0 loss to Roma.
Inter now trail Juventus and Napoli by five points while Roma, third last season, will bid for their second win on Monday against Atalanta.
"It's difficult to explain how we conceded those goals, but the reality is that the team is struggling with pressure," said Spalletti.
Meanwhile, surprise package SPAL, playing their second season in the top flight, are also riding high after their second win, a 1-0 victory over promoted Parma.
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