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Plasil had a glorious chance to break the deadlock just four minutes later when he received the ball inside the area, only to be denied again by Scuffet, who recovered in time to save from Gonzalo Bergessio moments later.
Udinese's first real chances of the match came with just six minutes left until the break when Antonio Di Natale had two efforts in quick succession, but on both occasions he was kept out by Mariano Andujar.
The hosts continued to threaten after the restart, coming close in the 51st minute when Bruno Fernandes sent his shot narrowly over the bar.
Fernandes came close again just before the hour mark, but this time his shot lacked power and was comfortable for Andujar to collect at his near post.
The deadlock was finally broken with just over 20 minutes remaining as Roberto Pereyra got behind the defence with a slick one-two before crossing for Di Natale to nod home from close range at the back post.
It was almost 2-0 in the 75th minute as Thomas Heurtaux burst into the box to latch on to a pass, but Andujar was equal to his effort.
The visitors threatened to get an equaliser with three chances in quick succession moments later, but Scuffet denied Fabian Monzon and Francesco Lodi, while Plasil's close-range header was well blocked.
Scuffet was called upon again with eight minutes remaining, but he was once more up the task as he kept out Bergessio's header with a fine stop.
The hosts held on in the closing stages for a victory that leaves Catania still six points from safety with seven matches to play.