Walter Mazzarri's side continued to keep pace at the top of Serie A, with a particular highlight being the return of Javier Zanetti
Bardi, on a season-long loan with the newly promoted side, was left red-faced when he directed Jonathan’s cross into his own goal in the first half.
The highly-rated 20-year-old, one of four Inter-owned players in Livorno’s matchday squad, was beaten again right at the death when Yuto Nagatomo sealed the points.
Bardi did his best to make up for his first-half error as he made two fine saves before Javier Zanetti made his comeback after an eight-month lay-off with an Achilles tendon injury.
An Inter win rarely looked in doubt and the victory ensured it remains in fourth place, six points behind leader Roma - which plays on Sunday.
Walter Mazzarri made one change to the Inter side that beat Udinese last time out, with Rolando replacing the suspended Juan Jesus.
Inter looked to dictate the play early on, but despite enjoying plenty of possession it struggled to fashion clear-cut opportunities against a well-drilled Livorno side.
The home side looked dangerous down the right flank and was gifted the lead on the half-hour mark when Jonathan's cross was inexplicably diverted into his own net by Bardi.
The young Italian made amends just before halftime when he palmed Ricky Alvarez’s effort wide to ensure Inter just had the one-goal lead at the break.
Livorno went in search of an equalizer early in the second half and Emerson tried his luck with a long-range free kick, but the Brazilian's driven effort flew just wide of Samir Handanovic's left-hand post.
Bardi kept Livorno in it just after the hour-mark as Nagatomo played in a low cross from the left, but Palacio's shot was kept out by the keeper from point-blank range.
Esteban Cambiasso went close with 16 minutes remaining as Inter strived for a two-goal cushion, but his low strike was well held by Bardi.
With nine minutes remaining Zanetti returned after initially planning to retire at the end of last season, and Inter completed the win in stoppage time when Nagatomo fired in after an excellent Mateo Kovacic run.