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 tribute to club president Massimo Moratti, who recently sold a controlling stake to Indonesian businessman Erick Thohir, but saved the loudest cheer of the night for Zanetti who came on with eight minutes left.
"It's really emotional. Making a comeback at 40-years-old isn't easy," said Zanetti, who has been sidelined since April with an Achilles tendon injury.
"I've said since the day of my operation I wanted at least one more game in front of my fans. I couldn't have asked for anything more."
Inter took the lead with a comical own goal. There seemed no danger when Jonathan whipped in a low cross that looked too close to Bardi to cause any problems but as the keeper stretched down to collect the ball he knocked it backwards into the net.
Zanetti, who came off the bench to rapturous applause, started the move which led to Inter's late second goal, feeding the ball to Kovacic who showed great skill in setting up Nagatomo.
The Japanese midfielder squeezed his shot past Bardi as he moved Inter to 29 goals for the season, equalling the club record after 12 matches set by the Ronaldo-era side and Jose Mourinho's treble-winning team.
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.