Paulinho and Innocent Emeghara scored second-half goals to cancel out efforts from Hernanes and Rodrigo Palacio in the first 45 minutes.
The result saw Livorno climb to within a point of safety, and coach Di Carlo - appointed to the role in January - wants to see his players show the same fight for the remainder of the season.
"It's a point that is worth so much for the team in terms of staying up," he is quoted as telling Sky Sport Italia. "We need to fight day after day to try and get the points we need. It wasn't easy to recover from two goals down.
"We have a team that is full of inexperience. We have a lot of young players and in some situations we have not responded well to conceding goals.
"Unfortunately our errors have been very costly and our opponents scored from the first real chance they created.
"The important thing is that this Livorno has character and a way of playing.
"We can always improve and I was pleased with the spirit of the team. That is the way to ensure we stay up."
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