Mazzarri's men looked to be in complete control at the Stadio Armando Picchi on Monday after goals from Hernanes and Rodrigo Palacio put them 2-0 up at half-time.
However, Livorno's top scorer Paulinho pulled one back early in the second half, before substitute Innocent Emeghara completed a dramatic comeback five minutes from time.
Emeghara slotted past Samir Handanovic after capitalising on a sloppy backpass from Fredy Guarin, leaving Mazzarri to lament his team's lack of focus.
"From the character point of view we have concentration limits," the former Napoli boss told Sky Sport.
"Unfortunately we have to work on this, I'm seeing some improvements, but it is necessary to concentrate for all 90 minutes of the match.
"It is crucial to always be on the ball during the match."
Meanwhile, Livorno coach Domenico Di Carlo praised the spirit his side displayed in moving just a point adrift of safety.
"It's a point that is worth so much for the team in terms of staying up," Di Carlo said.
"We can always improve and I was pleased with the spirit of the team. That is the way to ensure we stay up."
