Di Carlo replaced Attilio Perotti earlier this week and the change in command paid immediate dividends.
Goals from Leandro Greco, Paulinho and Marco Benassi handed the Amaranto a 3-0 lead after 26 minutes played.
Sassuolo's Domenico Berardi reduced the deficit before the half-hour mark from the penalty spot but the hosts held on to claim their fourth win of the campaign.
Livorno are 19th in the table after the win and sit just two points behind Bologna, who hold the last position of safety.
Sassuolo, meanwhile, are one place and one point better off than Livorno.
The home side had an excellent start with Greco netting from close range after latching onto a dangerous ball delivered by Marco Benassi in the fifth minute.
Paulinho doubled Livorno's lead six minutes later when he got past his marker and struck a left-footed shot from just outside the area towards the far post and out of Sassuolo goalkeeper Gianluca Pegolo's reach.
The hosts continued to surge forward and made it 3-0 when a counter-attack saw Innocent Emeghara feed Benassi in the box and his 12-yard attempt beat Pegolo.
In the 26th minute, the referee pointed to the penalty spot after Livorno goalkeeper Francesco Bardi brought down Nicola Sansone in the area.
Berardi struck the ball out of Bardi's reach to put his team back in the game.
Sassuolo had several chances to reduce the arrears in the second half.
In the 51st minute, Sansone's drive from 30 yards went just wide of the far post.
The visitors increased the tempo in the late stages of the game and Berardi's 10-yard strike went over the bar before Antonio Floro Flores' close-range header missed the target.