Walter Mazzarri's men went into the match at San Siro six points behind Juventus and Napoli, who play each other Sunday, and nine behind leaders Roma, who host struggling Sassuolo on Sunday.
And they managed to reduce the deficit on the sides occupying the UEFA Champions League qualification spots by condemning Livorno to their fourth straight top-flight away defeat.
The hosts' opening goal came in the 30th minute as Jonathan's right-wing cross was diverted into the net by Livorno goalkeeper Francesco Bardi - who is on loan at the Stadio Armando Picchi from Inter.
Club captain Javier Zanetti made his return to the fold after over eight months out with a snapped Achilles tendon and the Argentinian played a key role in the second goal that wrapped the three points up for Mazzarri's team.
The veteran picked out Mateo Kovacic, who embarked on a mazy run before neatly playing in Yuto Nagatomo, and the Japan international confidently fired past Bardi.
Saturday's early kick-off saw two out of form sides go head-to-head as Catania hosted Udinese, and the home side managed to pick up only their second win of the season to condemn the visitors to a fourth defeat in their last five games.
Maxi Lopez scored the goal's only game as he buried a penalty half-an-hour into the match, but the win was not enough to move Catania up the table.
Luigi Di Canio's men began the weekend in 19th place and although they have moved on to nine points - level with Sassuolo and Sampdoria - they remain second bottom.
