Francesco Guidolin's men showed no signs of nerves as they took the game straight to mid-table Catania, Di Natale firing wide in the first minute and Dusan Basta seeing his sixth-minute shot well saved by Juan Pablo Carrizo.
Udinese continued to pile on the pressure with Roberto Pereyra firing wide before Di Natale grabbed the opener in the 20th minute with an angled shot in the box after an assist from Maurizio Domizzi.
Alejandro Gomez had Catania's first chance in the 25th minute, and the hosts should have equalised in the 37th minute when Mehdi Benatia fouled Andrea Catellani in the box but Gomez fired the Catania spot-kick wide.
Di Natale went close again at the start of the second half and Udinese extended their lead in the 58th minute when Kwadwo Asamoah set up Fabbrini who hit a right-footed shot into the bottom left corner.
Gomez tried to atone for his penalty miss but his 60th-minute shot sailed straight into the arms of Udinese goalkeeper Samir Handanovic, then Catania substitute David Suazo also went close.
With the hosts enjoying their best spell of the game Nicolas Spolli had two good chances and Catellani brought a good save out of Handanovic before Udinese almost made it 3-0 in injury time when Asamoah hit the post from close range.