Inter Milan keeps pressure on leaders
Also, club captain Javier Zanetti made his comeback after snapping his Achilles tendon in April, a potentially career-ending injury.
''It's very emotional for me because it wasn't easy to come back after six months out,'' Zanetti said. ''After the operation I said I dreamed of playing at least one more match in front of our fans and I managed it. I want to thank everyone who helped me and was close to me during these difficult months. I couldn't ask for anything more.
''We have to stay focused to fight for something important. The table will tell us where we end up, but with this spirit we can go far.''
Inter took the lead on the half hour when Livorno goalkeeper Francesco Bardi dived to smother a Jonathan cross but somehow succeeded in pushing the ball into his own net.
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!''
Livorno remains 14th after its fifth defeat in seven matches.
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.