French defender Heurtaux slotted home from close range to give the visitors the lead for the second time in the game after Nicolas Lopez's early effort had been cancelled out by Luca Siligardi.
They held on to move seven points clear of their 17th-placed opponents, who are guaranteed to spend the festive period in the bottom three of Serie A.
Livorno had a real opportunity to move out of the relegation zone against a side that had won just one of their last six games and were still reeling from their first loss in 20 years against Torino at the Friuli stadium last time out.
While Siligardi was inches wide it was the Bianconeri who drew first blood in the 11th minute.
Livorno defender Emerson's poor clearance was pounced upon by Lopez, who drove into the box before drilling home.
The hosts responded strongly, with Leandro Greco, Andrea Luci and Siligardi forcing Zeljko Brkic into action, while Marco Biagianti shot narrowly wide.
Their persistence was rewarded in the 32nd minute when a marvellous pass from Paulinho allowed Silgardi to finally breach Brkic's goal.
Livorno bossed the rest of the half and although Luci, Pasquale Schiattarella and Federico Ceccherini went close, they nearly fell behind as Luci's deflection off Allan's cross forced goalkeeper Francesco Bardi to tip over.
Udinese enjoyed a decent start to the second half as Maurizio Domizzi and Andrea Lazzari missed the target from distance, while Bardi easily saved Dusan Basta's strike.
Francesco Guidolin's men kept pouring forward and it was no surprise when they re-took the lead in the 65th minute.
Antonio Di Natale's corner created havoc in the Livorno area and although Bardi managed to keep out Danilo's close-range effort, he was unable to stop Heurtaux's rebound.
That spurred Livorno into action although the closest they came to an equaliser was Paulinho's header over the bar, while the Brazilian forward also struck high over the bar late on.