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 they remain in fourth place, six points behind leaders Roma - who play on Sunday.
Walter Mazzarri made one change to the Inter side that beat Udinese last time out, with Rolando replacing the suspended Juan Jesus.
Livorno were without the suspended Andrea Coda, Leandro Rinaudo and Leandro Greco after the trio all picked up their fourth yellow cards of the season last weekend.
Inter looked to dictate the play early on, but despite enjoying plenty of possession they struggled to fashion clear-cut opportunities against a well drilled Livorno side.
The home side did have a chance to open the scoring after 21 minutes, though, as Saphir Taider crossed from the right but Jonathan headed well wide of the far post.
Inter looked dangerous down the right flank and they were 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 half-time 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 equaliser 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.
Inter got back into their stride and Walter Samuel could have doubled their lead when he rose highest to meet a corner, but his header comfortably cleared Bardi’s crossbar.
Bardi kept Livorno in it just after the hour-mark as Yuto 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.
