The Italy Under-21 goalkeeper is one of several Livorno players on loan from Inter.
Bardi atoned for his error with a good save to prevent Ricardo Alvarez from doubling Inter's lead in first-half stoppage time.
The 40-year-old Zanetti received a standing ovation when he came on for the final eight minutes.
And Zanetti started the move which led to Inter's second goal in stoppage time. The veteran defender went on a forward run before picking out Mateo Kovacic, who threaded the ball through for Yuto Nagatomo's shot. The whole team rushed to Zanetti to celebrate.
"Zanetti has something special and you can see why he's a world-class champion," Inter coach Walter Mazzarri said. "While the team was suffering, he came on and gave everyone a sense of calm. To come back at his age after such a long lay-off and be immediately so confident, it's extraordinary. He's bionic!"
Inter moved provisionally three points behind Juventus and Napoli, who play each other on Sunday.
Livorno remains 14th after its fifth defeat in seven matches.
Lowly Catania earned a valuable three points with a controversial 1-0 victory over Udinese.
Maxi Lopez scored the only goal from the penalty spot on the half hour after Maurizio Domizzi hauled down Nicola Legrottaglie as a corner kick was taken.
Udinese had two strong penalty appeals turned down. Panagiotis Tachtsidis appeared to touch a ball with his arm shortly after Catania's opener, and Luis Muriel's effort struck Legrottaglie's arm from point-blank range in the 69th.
Roberto Pereyra and Gianpiero Pinzi also hit the woodwork for Udinese but Catania held on for its first win under Luigi De Canio.