The Brazilian forward struck in the 66th and 72nd minutes before Bardi preserved the points to take Livorno into the top half of the table and leave Catania rooted to the foot of it with their third straight defeat this season.
The pressure therefore increases on Catania coach Rolando Maran, although he will wonder, like many, how his side failed to find the back of the net at the Stadio Armando Picchi.
A scrappy first half saw few chances, the best of which was a curling free-kick from Leandro Greco which only just missed the post before Innocent Emeghera was denied two minutes before the break by Nicolas Spolli, who blocked his shot on the goal line with his heel.
Livorno moved up a gear in the second half and Paulinho's shot on the turn was deflected onto the post by Pablo Barrientos.
Bardi made his first big save of the game in the 62nd minute to deny Gonzalo Bergessio, whose deflection on Jaroslav Plasil's shot wrong-footed the Livorno custodian.
Four minutes later, Paulinho gave the hosts the lead, slotting his shot under Mariano Andujar from next to the penalty spot after meeting Pasquale Schiattarella's low pass from the right.
Six minutes later, Andujar and Giuseppe Bellusci got in each other's way, presenting the ball to Emeghera, who pulled it back from the goal line to Paulinho. The Brazilian seemed to take an eternity before finding a gap to thread the ball between two defenders on the line.
Catania threw men forwards with little else to lose and they nearly scored when Bergessio got around the back, but his cross into the centre was turned behind by Bardi ahead of the approaching Barrientos.
Ten minutes from time, Bergessio hit the crossbar with an acrobatic overhead kick, summing up Catania's afternoon and their season so far as the ball rebounded to safety.