Marek Hamsik scored twice for the hosts, either side of half-time, after Jose Callejon had put Rafael Benitez's side ahead.
Inter had briefly gone top after defeating Genoa 2-0 at San Siro earlier in the day, with two goals in the final 15 minutes from Yuto Nagatomo and Rodrigo Palacio sealing the points for Walter Mazzarri's side.
Torino matched that result at home to newly-promoted Sassuolo, thanks to goals either side of the break from Matteo Brighi and Alessio Cerci.
Roma also ran out 2-0 winners as two goals in as many minutes midway through the second half from Daniele De Rossi and Alessandro Florenzi handed them victory away at Livorno.
Elsewhere, the Giallorossi's city rivals Lazio ran out 2-1 winners at home to Udinese.
Hernanes opened the scoring for the hosts after 13 minutes and Antonio Candreva doubled that advantage from the spot three minutes later. Substitute Luis Muriel's goal on the hour was not enough for Udinese to claim anything from the game.
Cagliari had to come from behind at home to Atalanta to get off the mark with a 2-1 victory. Guglielmo Stendardo struck first for the visitors after 27 minutes but that lead was cancelled out within a minute by Radja Nainggolan.
Matias Cabrera hit the winner after 63 minutes to hand all three points to Cagliari.
Parma and Cheivo played out a goalless draw in a low key start to the campaign for both sides.