Bardi patted a simple-looking Jonathan cross into his own net after half-an-hour before Nagatomo sealed the points in added time for Walter Mazzari's Inter, who welcomed back 40-year-old Javier Zanetti after seven months out injured.
Inter move on to 25 points but stay fourth six points adrift of unbeaten leaders AS Roma, who host Sassuolo on Sunday (1400 GMT), and three behind Napoli and Juventus, who meet also on Sunday in Turin (1945 GMT). Livorno remain 14th on 12 points.
The Inter fans paid an emotional tribute to president Massimo Moratti, who recently sold a controlling stake in the club to Indonesian businessman Erick Thohir.
A Maxi Lopez penalty in the 30th minute saw Catania beat Udinese 1-0 in a match that left the visitors furious about two spot kicks denied them after Panagiotis Tachtsidis and Nicola Legrottaglie both appeared to handle the ball.
The win was Catania's first under new manager Gigi De Canio and sees them move on to nine points, level with Sassuolo and Sampdoria, although they remain second-bottom in the table.
(Reporting by Terry Daley; Editing by Ken Ferris)
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