Guidolin threatened to step down after Sunday's 2-0 home defeat to Torino, their fourth loss in their past six Serie A games.
The poor run of form has dropped Udinese to 15th in the standings.
"I had a moment in which disappointment got the better of me," Guidolin told Gazzetta dello Sport.
"But I know full well I must be the first to urge the team to do better and ask the fans to stand by us.
"I have taken this team to qualify for Europe four times and in my career I have never had to fight for (Serie A) survival.
"However, this is the situation and we must do our best to get out of it.
"I was born a fighter and I always roll up my sleeves and get to work."
Udinese are just four points clear of the drop zone and end the calendar year at Livorno on Saturday.
"The next game with Livorno will be crucial," Guidolin said. "We need to be determined and hungry."
The 58-year-old Guidolin has been in charge of Udinese since leaving Palermo in the summer of 2010.