Fabrizio Miccoli put the Rosaneri in front after 10 minutes before two goals from Josip Ilicic put them in command.
Francesco Lodi pulled one back for Catania with 20 minutes remaining, but the points were never in any danger as Palermo pulled themselves away from the bottom three with a much-needed victory.
The derby had barely had time to warm up before Miccoli put the home side in front with his 100th goal in the Italian top flight.
He received the ball on the edge of the area and turned to shoot into the top-right corner, beyond the reach of goalkeeper Mariano Andujar.
Catania were already up against it and Palermo did not show them any mercy as they pushed forward repeatedly, going close to a second in the 22nd minute when Franco Brienza's shot from a tight angle struck the crossbar.
In the 38th minute, Miccoli sent a delightful ball over the top for Michel Morganella but he was put off as he was about to shoot by Nicolas Spolli, who timed his tackle to perfection when he risked giving away a penalty.
The only threat from Catania came from the well-known source of Lodi from a set-piece, but Francesco Benussi had done his homework and saved his curling free-kick.
Palermo needed just four minutes in the second half to double their lead with Brienza dribbling halfway around the perimeter of the penalty area, leaving three defenders in his wake before the ball fell to Ilicic to rifle in from 18 yards.
Ilicic scored his second on the hour mark when he carried the ball forward from inside his own half and, when the chance seemed to have gone, he turned inside onto his left foot and sent his shot inside the near post.
Ten minutes later, not even the best preparation from Benussi could keep out Lodi's next free-kick, which was curled over the wall to perfection.
Back at the other end, Brienza volleyed wide and Spolli lunged in the way of Morganella's shot towards an open goal, and the ball looped off the Catania defender over the bar.
Palermo teenager Paulo Dybala went close with two late chances, but no further goals were required to confirm a well-deserved home victory.